System Variation Of Constants Formula: Solve Forced Linear Systems

On this page: The Principle | Conditions | Misconceptions | EE Questions | Retrieval Practice | Worked Example | Solve a Problem | Related Principles | FAQ

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The Principle

Statement

For a nonhomogeneous first-order linear system,

$\mathbf{x}^{\prime}=A(t)\mathbf{x}+\mathbf{f}(t),$

a fundamental matrix $\Phi(t)$ for the associated homogeneous system $\mathbf{x}^{\prime}=A(t)\mathbf{x}$ lets you express the forced solution as

$\mathbf{x}(t)=\Phi(t)\left(\mathbf{c}+\int \Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right).$

The homogeneous flow $\Phi(t)\mathbf{c}$ carries the adjustable constants. The integral adds the forcing response in coordinates where the homogeneous motion has been factored out. In initial-value problems, the same idea is often written with limits from $t_0$ to $t$ so the initial vector can determine $\mathbf{c}$ cleanly.

Mathematical Form

$\mathbf{x}(t)=\Phi(t)\left(\mathbf{c}+\int \Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right)$

Where:

• $\mathbf{x}(t)$ = vector-valued solution of the nonhomogeneous system
• $A(t)$ = coefficient matrix in the linear system
• $\mathbf{f}(t)$ = forcing vector
• $\Phi(t)$ = invertible fundamental matrix for the associated homogeneous system
• $\mathbf{c}$ = arbitrary constant vector; initial conditions determine it once an antiderivative or base point is fixed

Initial-value form

For $\mathbf{x}(t_0)=\mathbf{x}_0$, the definite-integral form is

$\mathbf{x}(t)=\Phi(t)\left(\Phi(t_0)^{-1}\mathbf{x}_0+\int_{t_0}^{t}\Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right).$

When $\Phi(t_0)=I$, this simplifies to

$\mathbf{x}(t)=\Phi(t)\left(\mathbf{x}_0+\int_{t_0}^{t}\Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right).$

Why the inverse appears

Variation of constants starts from the homogeneous form $\mathbf{x}(t)=\Phi(t)\mathbf{u}(t)$ and lets the constant vector become a changing vector $\mathbf{u}(t)$. Differentiating gives

$\mathbf{x}^{\prime}=\Phi^{\prime}\mathbf{u}+\Phi\mathbf{u}^{\prime}.$

Because $\Phi^{\prime}=A\Phi$, the homogeneous part matches $A\mathbf{x}$ and the remaining term must supply the forcing:

$\Phi(t)\mathbf{u}^{\prime}(t)=\mathbf{f}(t).$

Multiplying by $\Phi(t)^{-1}$ gives $\mathbf{u}^{\prime}(t)=\Phi(t)^{-1}\mathbf{f}(t)$. Integrating this equation for $\mathbf{u}$ is what produces the integral inside the formula. This is the system-level analogue of variation of parameters, but the fundamental matrix handles all components at once.

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Conditions of Applicability

Condition: $\mathbf{x}^{\prime}=A\mathbf{x}+\mathbf{f}$; $\Phi^{\prime}=A\Phi$; $\Phi invertible$

Practical modeling notes

• The formula assumes you already have an invertible fundamental matrix for the associated homogeneous system.
• For a constant matrix $A$, $\Phi(t)=e^{At}$ is one common choice, but the formula itself is not limited to constant matrices.
• If a fundamental matrix is not available explicitly, the formula remains valid structurally, but it does not remove the need to solve or approximate the homogeneous system first.
• If using a definite integral from $t_0$ to $t$, keep the same base point when interpreting $\mathbf{c}$ and the initial condition.
• The forcing vector must be in the same system coordinates as $\mathbf{x}^{\prime}=A\mathbf{x}+\mathbf{f}$.

When It Doesn’t Apply

This formula is not a closed solution formula for every system:

• Nonlinear dependence on $\mathbf{x}$: the displayed explicit formula assumes the forcing is $\mathbf{f}(t)$, independent of the unknown solution. For $\mathbf{x}^{\prime}=A(t)\mathbf{x}+\mathbf{g}(t,\mathbf{x})$, the same substitution produces an implicit integral equation rather than this closed solution formula.
• Noninvertible matrix candidate: a matrix of solution columns is not usable here unless it is invertible on the working interval.
• Mismatched homogeneous system: $\Phi$ must solve the homogeneous system with the same coefficient matrix $A$ as the forced system.

Want the complete framework behind this guide? Read Masterful Learning.

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Common Misconceptions

Misconception 1: “The integrand is just the forcing vector”

The truth: the integrand is $\Phi(\tau)^{-1}\mathbf{f}(\tau)$, not only $\mathbf{f}(\tau)$.

Why this matters: the inverse fundamental matrix rewrites the forcing in coordinates aligned with the homogeneous flow.

Misconception 2: “Any matrix of convenient functions can be Phi”

The truth: $\Phi$ must satisfy $\Phi^{\prime}=A\Phi$ and be invertible.

Why this matters: if either check fails, the cancellation that leads to $\mathbf{u}^{\prime}=\Phi^{-1}\mathbf{f}$ fails.

Misconception 3: “Variation of constants replaces the homogeneous solution”

The truth: the formula uses the homogeneous solution matrix as its starting point.

Why this matters: the forced solution is built by combining homogeneous flow with accumulated forcing, not by ignoring the homogeneous system.

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Elaborative Encoding

Use these questions to build deep understanding. (See Elaborative Encoding for the full method.)

Within the Principle

• In $\mathbf{x}(t)=\Phi(t)\left(\mathbf{c}+\int \Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right)$, which part stores initial data, and which part stores accumulated forcing?
• Why does the formula need $\Phi^{-1}$ before integrating the forcing?

For the Principle

• Before using the formula, what two checks must you make about $\Phi$?
• How does choosing a lower integration limit change the way you interpret $\mathbf{c}$?

Between Principles

• How does this principle extend Matrix Exponential Solution Form from homogeneous systems to forced systems?

Generate an Example

• Describe a two-variable forced linear system where a diagonal fundamental matrix would make $\Phi^{-1}\mathbf{f}$ easy to compute.

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Retrieval Practice

Answer from memory, then click to reveal and check. (See Retrieval Practice for the full method.)

State the principle in words: _____A fundamental matrix converts a nonhomogeneous first-order linear system into an explicit variation-of-constants solution form.

Write the canonical equation: _____$\mathbf{x}(t)=\Phi(t)\left(\mathbf{c}+\int \Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right)$

State the canonical condition: _____$\mathbf{x}^{\prime}=A\mathbf{x}+\mathbf{f}; \Phi^{\prime}=A\Phi; \Phi invertible$

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Worked Example

Use this worked example to practice Self-Explanation.

Problem

For the forced system

$\mathbf{x}^{\prime}=\begin{bmatrix}1&0\\ 0&2\end{bmatrix}\mathbf{x}+\begin{bmatrix}e^t\\ e^{2t}\end{bmatrix},$

use the fundamental matrix

$\Phi(t)=\begin{bmatrix}e^t&0\\ 0&e^{2t}\end{bmatrix}$

and the initial condition $\mathbf{x}(0)=\begin{bmatrix}2\\ -1\end{bmatrix}$ to write $\mathbf{x}(t)$.

Step 1: Verbal Decoding

Target: $\mathbf{x}(t)$
Given: $A$, $\mathbf{f}(t)$, $\Phi(t)$, $\mathbf{x}(0)$
Constraints: first-order nonhomogeneous linear system; the supplied matrix is an invertible fundamental matrix for the matching homogeneous system; initial data are specified at time zero

Step 2: Visual Decoding

Draw two component lanes. Label the first lane with homogeneous factor $e^t$ and forcing $e^t$, and the second lane with homogeneous factor $e^{2t}$ and forcing $e^{2t}$. Mark $t=0$ as the base point where the initial vector enters. (Each forcing component becomes simple after multiplication by $\Phi^{-1}$.)

Step 3: Mathematical Modeling

1. $\Phi(t)=\begin{bmatrix}e^t&0\\ 0&e^{2t}\end{bmatrix},\qquad \mathbf{x}_0=\begin{bmatrix}2\\ -1\end{bmatrix}$
2. $\mathbf{f}(\tau)=\begin{bmatrix}e^\tau\\ e^{2\tau}\end{bmatrix}$
3. $\mathbf{x}(t)=\Phi(t)\left(\mathbf{x}_0+\int_0^t\Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right)$

Step 4: Mathematical Procedures

1. $\Phi(\tau)^{-1}=\begin{bmatrix}e^{-\tau}&0\\ 0&e^{-2\tau}\end{bmatrix}$
2. $\Phi(\tau)^{-1}\mathbf{f}(\tau)=\begin{bmatrix}1\\ 1\end{bmatrix}$
3. $\int_0^t \Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau=\begin{bmatrix}t\\ t\end{bmatrix}$
4. $\mathbf{x}(t)=\begin{bmatrix}e^t&0\\ 0&e^{2t}\end{bmatrix}\left(\begin{bmatrix}2\\ -1\end{bmatrix}+\begin{bmatrix}t\\ t\end{bmatrix}\right)$
5. $\underline{x_1(t)=e^t(t+2)}$
6. $\underline{x_2(t)=e^{2t}(t-1)}$

Step 5: Reflection

• Verification: differentiating gives $\mathbf{x}^{\prime}=\begin{bmatrix}e^t(t+3)\\ e^{2t}(2t-1)\end{bmatrix}$, which equals $A\mathbf{x}+\mathbf{f}(t)$.
• Initial-value check: substituting $t=0$ gives $\mathbf{x}(0)=\begin{bmatrix}2\\ -1\end{bmatrix}$.
• Connection to concept: multiplying by $\Phi^{-1}$ removes the homogeneous growth before the forcing is accumulated.

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Before moving on: self-explain the model

Try explaining Step 3 out loud or in writing: why $\Phi(0)=I$ makes $\Phi(0)^{-1}\mathbf{x}(0)=\mathbf{x}(0)$, why the forcing is multiplied by $\Phi^{-1}$, and why $\Phi(t)$ returns the accumulated result to the original coordinates.

Mathematical model with explanation

Principle: System Variation Of Constants Formula - $\mathbf{x}(t)=\Phi(t)\left(\mathbf{c}+\int \Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right)$.

Conditions: the system is first-order linear and nonhomogeneous, the given $\Phi$ satisfies $\Phi^{\prime}=A\Phi$, and $\Phi$ is invertible.

Relevance: the problem supplies the fundamental matrix, so the useful move is to use it to integrate the forcing rather than solve the coupled system from scratch.

Description: The inverse fundamental matrix changes the forcing into homogeneous-flow coordinates. The integral accumulates that changed forcing from the base point to $t$.

Goal: compute the pulled-back forcing integral and multiply by $\Phi(t)$ to obtain the initial-value solution.

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Solve a Problem

Apply what you’ve learned with Problem Solving.

Problem

For the forced upper-triangular system

$\mathbf{x}^{\prime}=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}\mathbf{x}+\begin{bmatrix}e^t(1+2t)\\ 2e^t\end{bmatrix},$

use

$\Phi(t)=\begin{bmatrix}e^t&te^t\\ 0&e^t\end{bmatrix}$

and $\mathbf{x}(0)=\begin{bmatrix}1\\ 0\end{bmatrix}$ to write $\mathbf{x}(t)$.

Hint (if needed): compute $\Phi(\tau)^{-1}\mathbf{f}(\tau)$ before integrating.

Show Solution

Step 1: Verbal Decoding

Target: $\mathbf{x}(t)$
Given: $A$, $\mathbf{f}(t)$, $\Phi(t)$, $\mathbf{x}(0)$
Constraints: first-order nonhomogeneous linear system; the supplied matrix is an invertible fundamental matrix for the matching homogeneous system; initial data are specified at time zero

Step 2: Visual Decoding

Draw a two-component vector lane where the second component feeds the first through the upper-right entry of the fundamental matrix. Mark the initial vector at $t=0$ and reserve one integral slot for each component. (The inverse fundamental matrix removes the coupled homogeneous flow before integrating.)

Step 3: Mathematical Modeling

1. $\Phi(t)=\begin{bmatrix}e^t&te^t\\ 0&e^t\end{bmatrix},\qquad \mathbf{x}_0=\begin{bmatrix}1\\ 0\end{bmatrix}$
2. $\mathbf{f}(\tau)=\begin{bmatrix}e^\tau(1+2\tau)\\ 2e^\tau\end{bmatrix}$
3. $\mathbf{x}(t)=\Phi(t)\left(\mathbf{x}_0+\int_0^t\Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right)$

Step 4: Mathematical Procedures

1. $\Phi(\tau)^{-1}=\begin{bmatrix}e^{-\tau}&-\tau e^{-\tau}\\ 0&e^{-\tau}\end{bmatrix}$
2. $\Phi(\tau)^{-1}\mathbf{f}(\tau)=\begin{bmatrix}1\\ 2\end{bmatrix}$
3. $\int_0^t \Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau=\begin{bmatrix}t\\ 2t\end{bmatrix}$
4. $\mathbf{x}(t)=\begin{bmatrix}e^t&te^t\\ 0&e^t\end{bmatrix}\begin{bmatrix}1+t\\ 2t\end{bmatrix}$
5. $\underline{x_1(t)=e^t(1+t+2t^2)}$
6. $\underline{x_2(t)=2te^t}$

Step 5: Reflection

• Verification: differentiating gives $\mathbf{x}^{\prime}=\begin{bmatrix}e^t(2+5t+2t^2)\\ 2e^t(1+t)\end{bmatrix}$, which equals $A\mathbf{x}+\mathbf{f}(t)$.
• Initial-value check: at $t=0$, the solution becomes $\begin{bmatrix}1\\ 0\end{bmatrix}$.
• Connection to concept: the upper-right entry in $\Phi$ shows why the forcing must be transformed before it is integrated.

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Related Principles

Principle	Relationship to System Variation Of Constants Formula
First-Order Linear System Form	Identifies the matrix-plus-forcing system that this formula solves after a fundamental matrix is known.
Matrix Exponential Solution Form	Supplies $\Phi(t)=e^{At}$ in constant homogeneous systems.
Nonhomogeneous Linear System Solution Structure	Explains how the forced solution combines a homogeneous family with one particular response.

See Differential Equations Subdomain for the full systems lane, and Principle Structures for organizing formulas, conditions, and neighboring principles.

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FAQ

What is the system variation-of-constants formula?

The system variation-of-constants formula is the solution form $\mathbf{x}(t)=\Phi(t)\left(\mathbf{c}+\int \Phi(\tau)^{-1}\mathbf{f}(\tau)\,d\tau\right)$ for a nonhomogeneous first-order linear system. It uses an invertible fundamental matrix from the associated homogeneous system.

When does the system variation-of-constants formula apply?

It applies when the system has the form $\mathbf{x}^{\prime}=A\mathbf{x}+\mathbf{f}$, the matrix $\Phi$ satisfies $\Phi^{\prime}=A\Phi$, and $\Phi$ is invertible on the working interval. The forcing must be the same forcing vector from the target system.

Why is Phi inverse in the formula?

$\Phi^{-1}$ converts the forcing vector into coordinates that remove the homogeneous flow. After that forcing is integrated, multiplying by $\Phi(t)$ returns the result to the original solution coordinates.

Is this the same as matrix exponential solution form?

No. Matrix Exponential Solution Form handles constant homogeneous systems with $\mathbf{x}(t)=e^{At}\mathbf{c}$. System variation of constants adds the forcing integral needed for nonhomogeneous systems.

Do I need a constant matrix A?

No. Constant matrices make one common case because $\Phi(t)=e^{At}$ is available. The formula itself uses any invertible fundamental matrix that satisfies $\Phi^{\prime}=A\Phi$.

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Related Guides

• Differential Equations Subdomain - Return to the systems and transforms map for this subdomain.
• Matrix Exponential Solution Form - Review the homogeneous flow that often supplies $\Phi$.
• Nonhomogeneous Linear System Solution Structure - Connect one forced response to the full solution family.
• Problem Solving - Practice turning a condition check into a solution plan.

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How This Fits in Unisium

Within the differential equations subdomain, Unisium treats System Variation Of Constants Formula as the bridge from homogeneous system flow to forced system response. The Unisium Study System pairs this guide with elaborative encoding, retrieval practice, and structured problem solving so you learn the stable sequence: verify $\Phi$, pull back the forcing, integrate, and push forward.

Ready to practice differential equations with structure? Check access and join the Unisium waitlist or explore the full framework in Masterful Learning.

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