Proof of the variation of constants formula for linear systems

Builds on System of linear differential equations Fundamental matrix Leibniz formula for determinants A matrix is invertible iff its determinant is a unit Inverse in terms of the adjugate The integral of a continuous function is an antiderivative Product rule Linearity of differentiation Extreme value theorem Cauchy–Schwarz inequality Lipschitz condition Uniqueness of solutions for Lipschitz vector fields

Proof

The fundamental matrix $\Phi$ is differentiable, so its entries are continuous on $I$ . By the Leibniz formula, $\det \Phi(s)$ is a polynomial in these entries, so it is continuous in $s$ .
Each $\Phi(s)$ is invertible, so $\det \Phi(s) \ne 0$ . Let $\Psi(s) = \Phi(s)^{-1}$ .
For $n = 1$ , $\Psi = 1/\Phi$ is continuous. For $n \ge 2$ , $\Psi(s) = \operatorname{adj} \Phi(s) / \det \Phi(s)$ by the adjugate formula. The entries of the adjugate are signed minors, which are polynomials in the entries of $\Phi$ . So $\Psi$ is continuous on $I$ .
Let $g(s) = \Psi(s) b(s)$ , a continuous map $I \to \R^n$ . Let $c(t) = \Psi(t_0) x_0 + \int_{t_0}^{t} g(s)\, ds$ , componentwise.
By the antiderivative theorem, applied to each component, $c$ is differentiable on $I$ with $c' = g$ .
Let $x(t) = \Phi(t) c(t)$ . This is the formula of the statement.
By the entrywise product rule and linearity of differentiation, $x' = \Phi' c + \Phi c' = A \Phi c + \Phi \Psi b = A x + b$ on $I$ .
Also $x(t_0) = \Phi(t_0) \Psi(t_0) x_0 + 0 = x_0$ . So $x$ solves the linear system $x' = A(t) x + b(t)$ on $I$ with $x(t_0) = x_0$ .
Uniqueness. Let $F(t, x) = A(t) x + b(t)$ on the open set $D = I \times \R^n$ . It is continuous.
Let $(t_1, x_1) \in D$ . Choose $[c, d] \subseteq I$ with $c < t_1 < d$ . By the extreme value theorem, the continuous function $t \mapsto \left(\sum_{i, j} A_{ij}(t)^2\right)^{1/2}$ has a maximum $K$ on $[c, d]$ .
For $t \in [c, d]$ and $v \in \R^n$ , the Cauchy–Schwarz inequality on each row gives $\norm{A(t) v} \le K \norm{v}$ . So $\norm{F(t, x) - F(t, y)} \le K \norm{x - y}$ on $(c, d) \times \R^n$ .
Thus $F$ is locally Lipschitz in $x$ . By uniqueness of solutions for Lipschitz vector fields, any solution on $I$ with value $x_0$ at $t_0$ equals $x$ .

Idea

Try $x = \Phi c$ with a varying vector $c$ . Since $\Phi' = A \Phi$ , the equation reduces to $\Phi c' = b$ , so $c' = \Phi^{-1} b$ . Integrating $c'$ from $t_0$ gives the formula, and uniqueness holds because the right side is locally Lipschitz in $x$ .

Reading path

To understand this page from the ground up, read these 550 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, 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 decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the formula for the derivative of a polynomial, Proof of the Laplace expansion along a column by transposition, 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 equivalence classes form a partition, 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 determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof that the exponential series has infinite radius of convergence, Proof that disjoint cycles commute, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a matrix is invertible iff its determinant is a unit, 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 that a matrix with two equal rows has determinant zero, 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 of the Laplace expansion along a row by the Leibniz formula, 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 that the determinant of a transpose equals the determinant, Proof of the laws of exponents in a group, Proof of the limit of a product of sequences, Proof of the inverse in terms of the adjugate, 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 that an intersection of subgroups is a subgroup, 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 that the sign is a homomorphism, 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 of the determinant of the inverse, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that the determinant of the identity matrix is one, 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 decomposition of a permutation into disjoint cycles, Proof of the reduction of a higher-order equation to a first-order system, Proof of the product of a matrix and its adjugate by Laplace expansion, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the division algorithm, Proof of Grönwall's inequality, Proof of the Archimedean property in an Archimedean ordered field, Proof of the sign of a cycle, Proof that finite-dimensional normed spaces are complete, Proof that a continuous real function on a compact set attains its bounds, Proof of the Leibniz formula from the defining properties of the determinant, Proof of continuous dependence on initial data by Grönwall's inequality, Proof of the uniqueness of inverses, 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 the determinant is multilinear in the rows, 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 subgroup criterion, 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.

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Equality of complex numbers statement
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Factorial formula for binomial coefficients statement
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System of linear differential equations definition
Uniqueness of inverses statement
Proof that a composition of continuous functions is continuous proof
Reduction of a higher-order equation to a first-order system statement
Universal set definition
Partition of a set definition
Homogeneous linear equation definition
Topological space definition
Order on the reals definition
Definite integral definition
Commutativity of addition statement
Tautology definition
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Proof of strong induction from ordinary induction proof
Uniform convergence definition
Sum of a finite geometric series statement
Upper and lower Darboux integrals definition
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Radius of convergence definition
Finite-dimensional vector space definition
Basis definition
Antisymmetric relation definition
Proof of the binomial theorem by induction proof
Reflexive relation definition
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Transitive relation definition
Picard operator definition
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Function definition
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Proof that a matrix with two equal rows has determinant zero proof
Proof that the determinant of the identity matrix is one proof
Uniform convergence on a set definition
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Laws of exponents in a group statement
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Metric space definition
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Determinant is multilinear in the rows statement
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Partial order definition
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Proof that disjoint cycles commute proof
Axiom of choice statement
Equivalence class definition
Intersection of subgroups is a subgroup statement
Epsilon-delta definition of continuity statement
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Double negation law statement
Well-ordering principle statement
Maximal element definition
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Composition of continuous functions is continuous statement
Proof that equivalence classes form a partition proof
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Proof by contradiction concept
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Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Proof that a function is bijective iff it has a two-sided inverse proof
Equivalence classes form a partition statement
Uniform continuity on a set definition
Open set definition
Division algorithm statement
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
Proof of the decomposition of a permutation into disjoint cycles proof
Hausdorff space definition
Open cover definition
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Continuous map of topological spaces definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Zorn's lemma statement
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The reals form an ordered field statement
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Decomposition of a permutation into disjoint cycles statement
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Open sets characterised by neighbourhoods statement
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Absolute value definition
Proof that the coordinate map is a linear isomorphism proof
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
Proof of the decomposition of a permutation into transpositions proof
Banach space definition
Continuity at a point of a topological space definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that closed sets contain their limits proof
Proof that open balls are open proof
Proof of the epsilon-N definition of the limit of a sequence proof
Every spanning set contains a basis statement
Isomorphism with the coordinate space statement
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Decomposition of a permutation into transpositions statement
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
Proof of continuity via closed sets proof
Closed sets contain their limits 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
Sign of a permutation definition
Modulus of a complex number definition
Compactness via open covers statement
Continuity via closed sets statement
Proof of the limit of a sum of sequences proof
Proof that the symmetric group is generated by transpositions proof
Intermediate value theorem statement
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof that differentiable functions are continuous 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 of the term test proof
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
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
The symmetric group is generated by transpositions statement
Proof that the sign is a homomorphism proof
Proof that a continuous image of a compact set is compact proof
Proof of the sign of a cycle proof
Lebesgue number lemma statement
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a metric space are bounded proof
Differentiable functions are continuous statement
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Linearity of the integral statement
Proof of the triangle inequality for norms proof
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Proof of the ratio formula for the radius of convergence proof
Completeness of the space of bounded continuous functions statement
Squeeze theorem statement
Convergent sequences are bounded statement
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
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Sign of a cycle statement
Compact subsets of a Hausdorff space are closed statement
Sign is a homomorphism statement
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Proof of the limit of a product of sequences proof
Proof that permuting rows multiplies the determinant by the sign 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 series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Permuting rows multiplies the determinant by the sign statement
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
Decreasing sequence bounded below converges statement
Geometric series example
Proof of Riemann's criterion for integrability proof
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
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Riemann's criterion for integrability statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Proof of the Cauchy–Hadamard theorem proof
Proof of the root test proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality 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 product rule proof
Proof of the linearity of differentiation proof
Leibniz formula for determinants statement
Cauchy product of absolutely convergent series statement
Bolzano–Weierstrass theorem statement
Proof of the additivity of the integral over intervals proof
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
Additivity of the integral over intervals statement
Ratio test statement
Proof of the Heine–Borel theorem by sequential compactness proof
Proof of the Laplace expansion along a row by the Leibniz formula proof
Root test statement
Proof that the determinant is multiplicative by expanding the rows proof
Proof that the determinant of a transpose equals the determinant proof
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
Heine–Borel theorem statement
Proof of the superposition principle proof
Proof of the power rule by induction proof
Boundedness theorem statement
Proof of the first fundamental theorem of calculus proof
Determinant is multiplicative statement
Laplace expansion along a row statement
Determinant of a transpose statement
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 determinant of the inverse proof
Proof of the Laplace expansion along a column by transposition proof
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
Determinant of the inverse statement
Proof of Fermat's theorem on stationary points proof
Laplace expansion along a column statement
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
Proof of the product of a matrix and its adjugate by Laplace expansion 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
Product of a matrix and its adjugate statement
Rolle's theorem statement
Proof that finite-dimensional normed spaces are complete proof
Proof of the mean value theorem for integrals proof
Proof of the inverse in terms of the adjugate proof
Proof of the Weierstrass M-test proof
Finite-dimensional normed spaces are complete statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Inverse in terms of the adjugate statement
Weierstrass M-test statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that a matrix is invertible iff its determinant is a unit proof
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
A matrix is invertible iff its determinant is a unit statement
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
Fundamental matrix definition
Proof of the variation of constants formula for linear systems proof

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Variation of constants formula for linear systems

Tag	`KAN26HZ6F2`
URI	`m2://KAN26HZ6F2`
Kind	proof
Writing	W0 stub
MSC	34A30
Proves	Variation of constants formula for linear systems
Uses	System of linear differential equations, Fundamental matrix, Leibniz formula for determinants, A matrix is invertible iff its determinant is a unit, Inverse in terms of the adjugate, The integral of a continuous function is an antiderivative, Product rule, Linearity of differentiation, Extreme value theorem, Cauchy–Schwarz inequality, Lipschitz condition, Uniqueness of solutions for Lipschitz vector fields
Changed	4 November 2026

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