Proof of the solutions for distinct real 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 The exponential function is its own derivative Chain rule The exponential function is positive Exponential of a sum Linearity of differentiation

Proof

Let $r \in \R$ and $e_r(x) = e^{r x}$ . By the chain rule and the derivative of the exponential, $e_r' = r\, e_r$ . By induction on $k$ , $e_r^{(k)} = r^k e_r$ for each $k \ge 0$ .
Write $L y = y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y$ . By step 1, $L e_r = \left(r^n + a_{n-1} r^{n-1} + \dots + a_0\right) e_r = p(r)\, e_r$ , where $p$ is the characteristic polynomial.
Since $p(r_k) = 0$ , each $e_{r_k}$ is a solution of the homogeneous equation with constant coefficients on $\R$ .
Claim: if $s_1, \dots, s_k$ are distinct reals, $c_1, \dots, c_k \in \R$ , and $\sum_j c_j e^{s_j x} = 0$ for all $x \in \R$ , then all $c_j = 0$ . We argue by induction on $k$ .
For $k = 1$ , $c_1 e^{s_1 x} = 0$ and $e^{s_1 x} > 0$ , so $c_1 = 0$ .
Let $k \ge 2$ and assume the claim for $k - 1$ exponents. Multiply the relation by $e^{-s_k x}$ . By the exponential of a sum, $\sum_{j < k} c_j e^{(s_j - s_k) x} + c_k = 0$ for all $x$ .
Differentiate, using step 1 and linearity of differentiation: $\sum_{j < k} c_j (s_j - s_k)\, e^{(s_j - s_k) x} = 0$ for all $x$ .
The numbers $s_j - s_k$ for $j < k$ are distinct. By the induction hypothesis, $c_j (s_j - s_k) = 0$ , and $s_j \ne s_k$ gives $c_j = 0$ for $j < k$ .
Then $c_k e^{s_k x} = 0$ , so $c_k = 0$ by step 5. This proves the claim.
By step 4, $e^{r_1 x}, \dots, e^{r_n x}$ are linearly independent in the space $S$ of solutions on $\R$ .
The constant coefficients are continuous on $\R$ . By the dimension theorem, $S$ has dimension $n$ .
By the basis criterion, $n$ linearly independent vectors in $S$ form a basis. So $\left(e^{r_1 x}, \dots, e^{r_n x}\right)$ is a fundamental system.
Each solution is a combination of a basis, so it has the form $c_1 e^{r_1 x} + \dots + c_n e^{r_n x}$ . Conversely each such combination lies in the vector space $S$ .

Idea

Each root $r$ gives a solution $e^{r x}$ , because differentiating $e^{r x}$ multiplies it by $r$ . Exponentials with different rates cannot cancel: dividing by one of them and differentiating removes it, and induction does the rest. Then $n$ independent solutions fill the $n$ -dimensional solution space.

Reading path

To understand this page from the ground up, read these 470 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 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 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 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 that the exponential series has infinite radius of convergence, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, 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 of the integral equation form of an initial value problem, Proof that the integral of a continuous function is an antiderivative, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, 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 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 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 of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, 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 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 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 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 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 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 that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, 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 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 open balls are open, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof of the superposition principle, 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 of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

Basis concept
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Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
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Mathematical induction concept
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Ordered field definition
Partition of an interval definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Uniform metric on bounded functions example
Vector space definition
p-metric on R^n example
Proof of Pascal's rule from the recursive definition proof
Real and imaginary parts definition
Set definition
Ordinary differential equation definition
Lipschitz condition definition
Ring definition
Ring of integers statement
Limit of a function definition
Proof that a linear map is injective iff its kernel is trivial proof
Local minimum definition
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Mesh of a partition definition
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Riemann sum definition
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Implication definition
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The reals form a field statement
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Pascal's rule statement
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Addition of natural numbers definition
Biconditional definition
Truth table concept
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Existential quantifier definition
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Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
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Zero is not a successor statement
Bounded sequence definition
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Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Inner product space definition
Linear independence definition
Kernel of a linear map definition
Linear isomorphism definition
Linear span definition
Completeness axiom statement
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Proof of the equality of complex numbers proof
Continuous function definition
Initial value problem definition
Proof of the reduction of a higher-order equation to a first-order system proof
Antiderivative definition
Differentiable function definition
One-sided derivative definition
Linear differential equation of order n definition
Union of a family of sets definition
Subset definition
Riemann integral definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Mathematical proof concept
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Propositional logic concept
Order on the natural numbers definition
A linear map is injective iff its kernel is trivial statement
Limit superior definition
The rationals form an ordered field statement
Proof of the sum of a finite geometric series proof
Convergent sequence definition
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Upper and lower Darboux sums definition
The sequence r^n tends to infinity for r > 1 example
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Partial sum definition
Power series definition
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Equality of complex numbers statement
Picard–Lindelöf hypotheses definition
Factorial formula for binomial coefficients statement
Proof that a composition of continuous functions is continuous proof
Binary relation definition
Reduction of a higher-order equation to a first-order system statement
System of linear differential equations definition
Universal set definition
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Definite integral definition
Order on the reals definition
Tautology definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Sum of a finite geometric series statement
Uniform convergence definition
Upper and lower Darboux integrals definition
Radius of convergence definition
Convergent series definition
Proof of the Steinitz exchange lemma by induction proof
Basis definition
Finite-dimensional vector space definition
Proof of the binomial theorem by induction proof
Picard operator definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Sequential compactness definition
Complement definition
Subspace topology definition
Characteristic equation definition
Function definition
Interval definition
Logical equivalence definition
Uniform convergence on a set definition
Strong induction statement
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Dimension definition
Series of functions definition
Absolute convergence definition
Sum of a series definition
Steinitz exchange lemma statement
Binomial theorem statement
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Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Codomain definition
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Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
Partial order definition
Epsilon-delta definition of continuity statement
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Left inverse definition
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Axiom of choice statement
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Linearly independent set of the right size is a basis statement
Maximal element definition
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Upper bound definition
Uniform continuity definition
Composition of continuous functions is continuous statement
Bijective function definition
Bounded set definition
Metric topology definition
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Proof by contradiction concept
Proof of the diameter characterisation of Cauchy sequences proof
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Uniform continuity on a set definition
Proof that a function is bijective iff it has a two-sided inverse proof
Open set definition
Epsilon characterisation of Cauchy sequences statement
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Hausdorff space definition
Closed set definition
Open cover definition
Continuous map of topological spaces definition
Equivalent metrics definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Complete metric space definition
Proof of the description of the open sets of a metric space proof
Zorn's lemma statement
The reals form an ordered field statement
Compact space definition
Proof that a set is open if and only if its complement is closed proof
Equivalent norms definition
Open sets characterised by neighbourhoods statement
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
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
Complement of an open set is closed statement
Compact set definition
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Continuity at a point of a topological space definition
Proof that closed sets contain their limits proof
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
Every spanning set contains a basis statement
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Closed sets contain their limits statement
Proof of continuity via closed sets 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 of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Compactness via open covers statement
Modulus of a complex number definition
Continuity via closed sets statement
Proof that differentiable functions are continuous proof
Proof of the limit of a sum of sequences proof
Intermediate value theorem statement
Proof of the uniqueness of limits proof
Proof of the squeeze theorem for sequences 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 of continuity via open sets in metric spaces proof
Proof that limits preserve non-strict inequalities proof
Proof that metric spaces are Hausdorff proof
Proof of the completeness of the space of bounded continuous functions proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
The sequence 1/n tends to zero example
Proof that Cauchy sequences are bounded proof
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the term test proof
Proof of the monotone convergence theorem for sequences proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
Differentiable functions are continuous statement
Epsilon-delta continuity in metric spaces statement
Lebesgue number lemma statement
Proof that compact subsets of a metric space are bounded proof
Proof that compactness and sequential compactness agree in metric spaces proof
Metric spaces are Hausdorff statement
Limit of a sum of sequences statement
Linearity of the integral statement
Uniqueness of limits statement
Proof of the monotonicity of the integral proof
Proof of the triangle inequality for norms proof
Limits preserve non-strict inequalities statement
Completeness of the space of bounded continuous functions statement
Proof of the ratio formula for the radius of convergence proof
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Compact subsets of a Hausdorff space are closed statement
A complex number times its conjugate statement
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Compact subsets of a metric space are bounded statement
Continuity via open sets statement
Proof of the limit of a product of sequences proof
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Term test statement
Triangle inequality for norms 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
Proof of the Banach fixed point theorem by iteration proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Proof of the Cauchy–Schwarz inequality proof
Limit of a product of sequences statement
Proof of Riemann's criterion for integrability proof
Decreasing sequence bounded below converges statement
Geometric series example
A series of nonnegative terms converges iff its partial sums are bounded statement
Banach fixed point theorem statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Riemann's criterion for integrability statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Proof of the root test proof
Proof of the Cauchy–Hadamard theorem proof
Proof of the ratio test proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Cauchy–Schwarz inequality statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof that an iterate of the Picard operator is a contraction proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Comparison test statement
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
An iterate of the Picard operator is a contraction statement
Chain rule statement
Product rule statement
Linearity of differentiation statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Proof of the Heine–Borel theorem by sequential compactness proof
Additivity of the integral over intervals statement
Ratio test 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
Proof of the constant multiple rule proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the power rule by induction proof
Heine–Borel theorem statement
Boundedness theorem statement
Proof of the superposition principle proof
Proof of the first fundamental theorem of calculus proof
Cauchy–Hadamard theorem statement
Ratio formula for the radius of convergence statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Power rule statement
Superposition principle statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Proof that a continuous real function on a compact set attains its bounds proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that continuous functions are Riemann integrable proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere 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
Proof of Rolle's theorem proof
Absolutely convergent series converge statement
Rolle's theorem statement
Proof of the mean value theorem for integrals proof
Proof that finite-dimensional normed spaces are complete proof
Proof of the Weierstrass M-test proof
Proof of the mean value theorem from Rolle's theorem proof
Finite-dimensional normed spaces are complete statement
Mean value theorem for integrals statement
Weierstrass M-test statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
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
The integral of a continuous function is an antiderivative statement
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
Proof of term-by-term differentiation proof
The exponential series has infinite radius of convergence statement
Fundamental theorem of calculus statement
Term-by-term differentiation statement
Exponential function definition
Proof of the integral equation form of an initial value problem proof
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
Exponential of a sum statement
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 the Picard–Lindelöf theorem proof
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
The exponential function is positive statement
Picard–Lindelöf theorem statement
The exponential function is its own derivative statement
Proof of Grönwall's inequality proof
Grönwall's inequality statement
Uniqueness of solutions for Lipschitz vector fields statement
Proof of continuous dependence on initial data by Grönwall's inequality proof
Continuous dependence on initial data statement
Maximal solution definition
Proof of global existence for globally Lipschitz vector fields by steps of uniform length proof
Global existence for globally Lipschitz vector fields statement
Proof that the solution space of a homogeneous linear equation has dimension n proof
Solution space of a homogeneous linear equation has dimension n statement
Fundamental system of solutions definition
Proof of the solutions for distinct real roots proof

Tag	`R8VNSQBR4V`
URI	`m2://R8VNSQBR4V`
Kind	proof
Writing	W0 stub
MSC	34A30
Proves	Solutions for distinct real 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, The exponential function is its own derivative, Chain rule, The exponential function is positive, Exponential of a sum, Linearity of differentiation
Changed	4 November 2026

Proof of the solutions for distinct real roots

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