Proof of the solutions for repeated roots

Builds on Characteristic equation Linear equation with constant coefficients Fundamental system of solutions Solution space of a homogeneous linear equation has dimension n Linearly independent set of the right size is a basis Linear independence Multiplicity of a root Fundamental theorem of algebra Factor theorem A polynomial over a field has at most as many roots as its degree R[x] is an integral domain iff R is Cancellation in an integral domain Derivative of a polynomial The exponential function is its own derivative Chain rule Product rule Linearity of differentiation The exponential function is positive Exponential of a sum

Proof

We factor the operator through $(D - r)^m$ .

Let $D$ be differentiation on the infinitely differentiable functions $\R \to \R$ . For a real polynomial $q(t) = \sum_i b_i t^i$ , let $q(D) u = \sum_i b_i u^{(i)}$ . The equation is $p(D) y = 0$ , with $p$ the characteristic polynomial.
For real polynomials $q_1, q_2$ , we have $q_1(D)\left(q_2(D) u\right) = (q_1 q_2)(D) u$ . Indeed, $D^i$ is linear by linearity of differentiation and $D^i D^j = D^{i+j}$ . Expanding both sides gives the same sum.
Let $s \in \R$ and $k \ge 0$ . By the product rule, the chain rule and the derivative of the exponential, $D\left(x^k e^{s x}\right) = k x^{k-1} e^{s x} + s\, x^k e^{s x}$ . So $(D - s)\left(x^k e^{s x}\right) = k x^{k-1} e^{s x}$ , which is $0$ for $k = 0$ .
Applying step 3 $j$ times, $(D - s)^j \left(x^k e^{s x}\right)$ is a constant multiple of $x^{k-j} e^{s x}$ for $j \le k$ . It is $0$ for $j = k + 1$ , and so for each $j > k$ .
Solutions. Since $r$ has multiplicity $m$ , $p(t) = h(t) (t - r)^m$ for a real polynomial $h$ . Let $0 \le k \le m - 1$ .
By steps 2 and 4, $p(D)\left(x^k e^{r x}\right) = h(D)\left((D - r)^m \left(x^k e^{r x}\right)\right) = h(D)\, 0 = 0$ . So $x^k e^{r x}$ solves the equation with constant coefficients on $\R$ .
Lemma. Let $s_1, \dots, s_k$ be distinct reals and $q_1, \dots, q_k$ real polynomials with $\sum_j q_j(x) e^{s_j x} = 0$ for all $x \in \R$ . Then each $q_j$ is the zero polynomial. We argue by induction on $k$ .
For $k = 1$ , $e^{s_1 x} > 0$ , so $q_1(x) = 0$ for every $x \in \R$ . A nonzero polynomial has finitely many roots, so $q_1 = 0$ .
Let $k \ge 2$ . Multiply by $e^{-s_k x}$ and use the exponential of a sum: $\sum_{j < k} q_j(x) e^{\mu_j x} + q_k(x) = 0$ , where $\mu_j = s_j - s_k \ne 0$ are distinct.
For a polynomial $q$ and $\mu \ne 0$ , step 3 gives $D\left(q(x) e^{\mu x}\right) = \left(q' + \mu q\right)(x)\, e^{\mu x}$ . If $q \ne 0$ has degree $d$ and leading coefficient $a$ , then $q' + \mu q$ has degree $d$ and leading coefficient $\mu a \ne 0$ , by the derivative of a polynomial.
Let $e$ be one more than the degree of $q_k$ , or $e = 0$ if $q_k = 0$ . Differentiating the relation of step 9 $e$ times kills $q_k$ . By step 10, it gives $\sum_{j < k} \tilde q_j(x) e^{\mu_j x} = 0$ with $\tilde q_j \ne 0$ whenever $q_j \ne 0$ .
By the induction hypothesis, each $\tilde q_j = 0$ , so each $q_j = 0$ for $j < k$ . Then $q_k(x) e^{s_k x} = 0$ , and $q_k = 0$ by step 8.
Independence. If $\sum_{k=0}^{m-1} c_k x^k e^{r x} = 0$ on $\R$ , the lemma with one exponent gives $\sum_k c_k x^k = 0$ as a polynomial. So each $c_k = 0$ , and the $m$ functions are linearly independent.
Now assume all roots of $p$ are real. We show $p(t) = \prod_{i=1}^{n} (t - c_i)$ with real $c_i$ , by induction on $n$ . For $n = 0$ the product is empty.
For $n \ge 1$ , the fundamental theorem of algebra gives a root $c$ of $p$ , real by assumption. By the factor theorem in $\R[t]$ , $p = (t - c)\, p_1$ with $p_1$ real and monic of degree $n - 1$ .
Each complex root $z$ of $p_1$ is a root of $p$ , since $p(z) = (z - c) p_1(z)$ . So it is real, and the induction hypothesis applies to $p_1$ .
Let $\rho_1, \dots, \rho_s$ be the distinct values among the $c_i$ , with $\rho_j$ occurring $m_j$ times. Then $p(t) = \prod_j (t - \rho_j)^{m_j}$ and $m_1 + \dots + m_s = n$ . Each root $z$ of $p$ makes some factor vanish, so the roots are exactly $\rho_1, \dots, \rho_s$ .
The multiplicity of $\rho_j$ is $m_j$ . It is at least $m_j$ by step 17. Suppose $(t - \rho_j)^{m_j + 1}$ divides $p$ , say $p = (t - \rho_j)^{m_j + 1} g$ .
Since [[ $\R[t]$ is an integral domain|0B5Z1EC66J]], we may cancel $(t - \rho_j)^{m_j}$ . This gives $\prod_{i \ne j} (t - \rho_i)^{m_i} = (t - \rho_j)\, g$ . At $t = \rho_j$ the left side is nonzero and the right side is $0$ , a contradiction.
So the families for the distinct roots have $m_1 + \dots + m_s = n$ functions in total, all solutions by step 6.
They are linearly independent. A vanishing combination can be grouped as $\sum_j q_j(x) e^{\rho_j x} = 0$ , where the coefficients of $q_j$ are the coefficients used for $\rho_j$ . By the lemma each $q_j = 0$ , so all coefficients vanish.
The space $S$ of solutions on $\R$ has dimension $n$ , by the dimension theorem. By the basis criterion, these $n$ independent solutions form a basis, that is, a fundamental system.

Idea

The operator $p(D)$ factors like the polynomial $p$ , and the factor $(D - r)$ lowers the power of $x$ in $x^k e^{r x}$ by one. So $(D - r)^m$ kills $x^k e^{r x}$ for $k < m$ . Independence follows because differentiating a polynomial times an exponential keeps the degree, while differentiating a polynomial alone lowers it.

Reading path

To understand this page from the ground up, read these 596 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Steinitz exchange lemma by induction, Proof of uniqueness of solutions for Lipschitz vector fields, Proof of the Cauchy–Hadamard theorem, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of cancellation in an integral domain, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the completeness of the space of bounded continuous functions, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the fundamental theorem of algebra by Liouville's theorem, Proof that the solution space of a homogeneous linear equation has dimension n, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of the reverse triangle inequality, Proof that the exponential series has infinite radius of convergence, Proof of the derivative as a little-o approximation, Proof that R[x] is an integral domain iff R is, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function with nonnegative derivative is increasing, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of continuity via closed sets, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that the exponential function is positive, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, Proof of the integral equation form of an initial value problem, Proof that the integral of a continuous function is an antiderivative, Proof of the degree of a product over an integral domain, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof of Cauchy's estimates, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the Cauchy–Riemann equations, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof of the chain rule for a composition with a curve, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof of Cauchy's integral formula for derivatives, Proof of the description of the open sets of a metric space, Proof of the Banach fixed point theorem by iteration, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that a zero derivative on a connected open set implies constant, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the modulus is multiplicative, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that a polynomial over a field has at most as many roots as its degree, Proof of the remainder theorem, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that a linear map is injective iff its kernel is trivial, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof of integration by substitution, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that an iterate of the Picard operator is a contraction, Proof of the second fundamental theorem of calculus, Proof of global existence for globally Lipschitz vector fields by steps of uniform length, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the reduction of a higher-order equation to a first-order system, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof that n independent or n spanning vectors in an n-dimensional space form a basis, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the fundamental theorem of contour integration, Proof of Goursat's theorem by repeated subdivision, Proof of Grönwall's inequality, Proof of the Archimedean property in an Archimedean ordered field, Proof that finite-dimensional normed spaces are complete, Proof that a continuous real function on a compact set attains its bounds, Proof of continuous dependence on initial data by Grönwall's inequality, Proof of the derivative of a map into a product, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that absolutely convergent series in a Banach space converge, Proof that open balls are open, Proof that power series are holomorphic in the disc of convergence, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the factor theorem, Proof that Lipschitz maps are uniformly continuous, Proof of the ratio formula for the radius of convergence, Proof of the superposition principle, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the Picard–Lindelöf theorem, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that sums and products of continuous functions are continuous, Proof of Liouville's theorem by Cauchy's estimates, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof of the constant multiple rule.

Basis concept
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Chain definition
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Bijective iff it has a two-sided inverse statement
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Lipschitz functions are uniformly continuous statement
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Proof that a set is open if and only if its complement is closed proof
Proof that the composition of continuous maps is continuous proof
Open sets of a metric space statement
Compact space definition
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Open sets characterised by neighbourhoods statement
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Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
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Complement of an open set is closed statement
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Banach space definition
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Proof that closed sets contain their limits proof
Compact set definition
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Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Proof of the epsilon-N definition of the limit of a sequence proof
Isomorphism with the coordinate space statement
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Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
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Proof of continuity via closed sets proof
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Triangle inequality statement
Absolute value is multiplicative statement
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the intermediate value theorem by the supremum proof
Continuity via closed sets statement
Modulus of a complex number definition
Proof of the limit of a sum of sequences proof
Proof of the reverse triangle inequality proof
Proof of the uniqueness of limits proof
Compactness via open covers statement
Proof of the squeeze theorem for sequences proof
Proof of continuity via open sets in metric spaces proof
Proof that metric spaces are Hausdorff proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that limits preserve non-strict inequalities proof
Proof of the completeness of the space of bounded continuous functions proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
The sequence 1/n tends to zero example
Proof that Cauchy sequences are bounded proof
Proof of the term test proof
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Reverse triangle inequality statement
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof of Cantor's intersection theorem proof
Proof that a continuous image of a compact set is compact proof
Metric spaces are Hausdorff statement
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a metric space are bounded proof
Lebesgue number lemma statement
Proof that compactness and sequential compactness agree in metric spaces proof
Proof of the sequence characterisation of uniform continuity proof
Limit of a sum of sequences statement
Proof of the triangle inequality for norms proof
Completeness of the space of bounded continuous functions statement
Uniqueness of limits statement
Differentiable functions are continuous statement
Limits preserve non-strict inequalities statement
Linearity of the integral statement
Proof of the ratio formula for the radius of convergence proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Proof of the monotonicity of the integral proof
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
A complex number times its conjugate statement
Compact subsets of a Hausdorff space are closed statement
Proof of the Cauchy–Riemann equations proof
Cantor's intersection theorem statement
Continuous image of a compact set is compact statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Compactness and sequential compactness agree in metric spaces statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
There is a natural number n with 1/n below any positive epsilon statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
Triangle inequality for norms statement
Term test statement
Proof that a decreasing sequence bounded below converges proof
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
The sequence r^n tends to zero for |r| < 1 example
Entire function definition
Cauchy–Riemann equations statement
Proof that the modulus is multiplicative proof
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof that absolutely convergent series in a Banach space converge proof
Proof that Lipschitz maps are uniformly continuous proof
Proof of the Banach fixed point theorem by iteration proof
Proof of the sequential characterization of limits proof
Proof of the Cauchy–Schwarz inequality proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Proof of Riemann's criterion for integrability proof
Geometric series example
A series of nonnegative terms converges iff its partial sums are bounded statement
Modulus is multiplicative statement
Absolutely convergent series in a Banach space converge statement
Banach fixed point theorem statement
Lipschitz maps are uniformly continuous statement
Riemann's criterion for integrability statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the root test proof
Proof of the ratio test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Proof that an iterate of the Picard operator is a contraction proof
Proof of the chain rule with an auxiliary difference quotient proof
Comparison test statement
Proof of the linearity of differentiation proof
Proof of the product rule proof
Proof that sums and products of continuous functions are continuous proof
Cauchy product of absolutely convergent series statement
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Proof of the derivative of a map into a product proof
An iterate of the Picard operator is a contraction statement
Product of continuous functions is continuous statement
Operator norm definition
Chain rule statement
Proof of the Heine–Borel theorem by sequential compactness proof
Linearity of differentiation statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Ratio test statement
Additivity of the integral over intervals statement
Product rule statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Completeness of the real numbers statement
Derivative of a map into a product statement
Proof of the superposition principle proof
Heine–Borel theorem statement
Proof of the constant multiple rule proof
Proof that power series are holomorphic in the disc of convergence proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Proof of the power rule by induction proof
Proof of the first fundamental theorem of calculus proof
Ratio formula for the radius of convergence statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Power series are holomorphic in the disc of convergence statement
Superposition principle statement
Constant multiple rule statement
Derivative of a map into R^n by components statement
Power rule statement
Proof that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of the formula for the derivative of a polynomial proof
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Proof that continuous functions are Riemann integrable proof
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Equivalence of norms in finite dimension statement
Absolutely convergent series converge statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof of the estimation lemma proof
Proof that finite-dimensional normed spaces are complete proof
Proof of the mean value theorem for integrals proof
Proof of the Weierstrass M-test proof
Linear maps on finite-dimensional spaces are continuous statement
Estimation lemma statement
Finite-dimensional normed spaces are complete statement
Mean value theorem for integrals statement
Weierstrass M-test statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem statement
First fundamental theorem of calculus statement
Proof of the chain rule in several variables proof
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Proof that a function with nonnegative derivative is increasing proof
Proof that the integral of a continuous function is an antiderivative proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof of the second fundamental theorem of calculus proof
A function with nonnegative derivative is increasing statement
Continuous partial derivatives imply differentiability statement
The integral of a continuous function is an antiderivative statement
Proof of the chain rule for a composition with a curve proof
Uniform convergence of derivatives allows term-by-term differentiation statement
Second fundamental theorem of calculus statement
Proof that the exponential series has infinite radius of convergence proof
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof of integration by substitution proof
The exponential series has infinite radius of convergence statement
Proof of term-by-term differentiation proof
Proof of the mean value inequality proof
Integration by substitution statement
Fundamental theorem of calculus statement
Term-by-term differentiation statement
Exponential function definition
Proof of the fundamental theorem of contour integration proof
Proof of the integral equation form of an initial value problem proof
Mean value inequality statement
Proof of the exponential of a sum by the Cauchy product proof
Proof of uniqueness of solutions for Lipschitz vector fields proof
Proof that power series are differentiable inside the radius of convergence proof
Fundamental theorem of contour integration statement
Exponential of a sum statement
Proof that a zero derivative on a connected open set implies constant proof
Integral equation form of an initial value problem statement
Power series are differentiable inside the radius of convergence statement
Proof that the exponential function is positive proof
Proof of Goursat's theorem by repeated subdivision proof
Zero derivative on a connected open set implies constant statement
Proof of the Picard–Lindelöf theorem proof
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
Goursat's theorem statement
The exponential function is positive statement
Picard–Lindelöf theorem statement
The exponential function is its own derivative statement
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Proof of Grönwall's inequality proof
Cauchy's integral formula statement
Grönwall's inequality statement
Proof that holomorphic functions are analytic proof
Uniqueness of solutions for Lipschitz vector fields statement
Proof of continuous dependence on initial data by Grönwall's inequality proof
Holomorphic functions are analytic statement
Continuous dependence on initial data statement
Maximal solution definition
Proof that holomorphic functions are infinitely differentiable proof
Proof of global existence for globally Lipschitz vector fields by steps of uniform length proof
Holomorphic functions are infinitely differentiable statement
Global existence for globally Lipschitz vector fields statement
Proof of Cauchy's integral formula for derivatives proof
Proof that the solution space of a homogeneous linear equation has dimension n proof
Cauchy's integral formula for derivatives statement
Solution space of a homogeneous linear equation has dimension n statement
Proof of Cauchy's estimates proof
Fundamental system of solutions definition
Cauchy's estimates statement
Proof of Liouville's theorem by Cauchy's estimates proof
Liouville's theorem statement
Proof of the fundamental theorem of algebra by Liouville's theorem proof
Fundamental theorem of algebra statement
Proof of the solutions for repeated roots proof

Tag	`R1R6024PCZ`
URI	`m2://R1R6024PCZ`
Kind	proof
Writing	W0 stub
MSC	34A30
Proves	Solutions for repeated roots
Uses	Characteristic equation, Linear equation with constant coefficients, Fundamental system of solutions, Solution space of a homogeneous linear equation has dimension n, Linearly independent set of the right size is a basis, Linear independence, Multiplicity of a root, Fundamental theorem of algebra, Factor theorem, A polynomial over a field has at most as many roots as its degree, R[x] is an integral domain iff R is, Cancellation in an integral domain, Derivative of a polynomial, The exponential function is its own derivative, Chain rule, Product rule, Linearity of differentiation, The exponential function is positive, Exponential of a sum
Changed	4 November 2026

Proof of the solutions for repeated roots

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