Linear System of ODEs Solver (3x3)

Published on: July 18, 2026
Final Answer: Free • Full Steps: Pro

This Linear System ODE Solver solves a 3×3 first-order system x' = A x using the eigenvalue method. It finds the eigenvalues and eigenvectors of the coefficient matrix A and combines them into the general solution.

Step-by-step method

  1. Write the system in matrix form x' = A x.
  2. Use the eigenvalue method: x(t) is a combination of e^(λ t) v.
  3. Find the eigenvalues of A.
  4. Find an eigenvector for each eigenvalue.
  5. Combine them into the general solution.

Formula:

\(\vec{x}(t) = C_{1} e^{\lambda_{1} t} \vec{v}_{1} + C_{2} e^{\lambda_{2} t} \vec{v}_{2} + \cdots\)

Example 1:

\\( ec{x}\,' = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{pmatrix} ec{x}\\)

Step 1 - Write the system in matrix form x' = A x.

In this problem: The system in matrix form is \( ec{x}\,' = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{pmatrix} ec{x}\), with \(A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{pmatrix}\).

\( ec{x}\,' = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{pmatrix} ec{x}\)

Step 2 - Use the eigenvalue method: x(t) is a combination of e^(λ t) v.

In this problem: Each eigenvalue and eigenvector pair contributes one term to the solution.

\(\vec{x}(t) = C_{1} e^{\lambda_{1} t} \vec{v}_{1} + C_{2} e^{\lambda_{2} t} \vec{v}_{2} + \cdots\)

Step 3 - Find the eigenvalues of A.

In this problem: Solving det(A - \(\lambda\) I) = 0 gives \(\lambda = 1,\; \lambda = 2,\; \lambda = 3\).

\(\lambda = 1,\; \lambda = 2,\; \lambda = 3\)

Step 4a - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 1\), an eigenvector is \(\left[\begin{matrix}2\\-2\\1\end{matrix}\right]\).

\(\lambda = 1:\; ec{v} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}\)

Step 4b - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 2\), an eigenvector is \(\left[\begin{matrix}0\\-1\\1\end{matrix}\right]\).

\(\lambda = 2:\; ec{v} = \begin{pmatrix} 0 \\ -1 \\ 1 \end{pmatrix}\)

Step 4c - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 3\), an eigenvector is \(\left[\begin{matrix}0\\0\\1\end{matrix}\right]\).

\(\lambda = 3:\; ec{v} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\)

Step 5 - Combine them into the general solution.

In this problem: Combine the eigenvalue terms into the general solution with constants C1, C2, and so on.

\(egin{gathered} x_{1}(t) = 2 C_{1} e^{t} \\ x_{2}(t) = \left(- 2 C_{1} - C_{2} e^{t}\right) e^{t} \\ x_{3}(t) = \left(C_{1} + C_{2} e^{t} + C_{3} e^{2 t}\right) e^{t} \end{gathered}\)

Final answer:

\\(egin{gathered} x_{1}(t) = 2 C_{1} e^{t} \\ x_{2}(t) = \left(- 2 C_{1} - C_{2} e^{t}\right) e^{t} \\ x_{3}(t) = \left(C_{1} + C_{2} e^{t} + C_{3} e^{2 t}\right) e^{t} \end{gathered}\\)

Example 2:

\\( ec{x}\,' = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix} ec{x}\\)

Step 1 - Write the system in matrix form x' = A x.

In this problem: The system in matrix form is \( ec{x}\,' = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix} ec{x}\), with \(A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix}\).

\( ec{x}\,' = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix} ec{x}\)

Step 2 - Use the eigenvalue method: x(t) is a combination of e^(λ t) v.

In this problem: Each eigenvalue and eigenvector pair contributes one term to the solution.

\(\vec{x}(t) = C_{1} e^{\lambda_{1} t} \vec{v}_{1} + C_{2} e^{\lambda_{2} t} \vec{v}_{2} + \cdots\)

Step 3 - Find the eigenvalues of A.

In this problem: Solving det(A - \(\lambda\) I) = 0 gives \(\lambda = -1,\; \lambda = 2,\; \lambda = 3\).

\(\lambda = -1,\; \lambda = 2,\; \lambda = 3\)

Step 4a - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = -1\), an eigenvector is \(\left[\begin{matrix}0\\0\\1\end{matrix}\right]\).

\(\lambda = -1:\; ec{v} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\)

Step 4b - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 2\), an eigenvector is \(\left[\begin{matrix}1\\0\\0\end{matrix}\right]\).

\(\lambda = 2:\; ec{v} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\)

Step 4c - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 3\), an eigenvector is \(\left[\begin{matrix}0\\1\\0\end{matrix}\right]\).

\(\lambda = 3:\; ec{v} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\)

Step 5 - Combine them into the general solution.

In this problem: Combine the eigenvalue terms into the general solution with constants C1, C2, and so on.

\(egin{gathered} x_{1}(t) = C_{1} e^{2 t} \\ x_{2}(t) = C_{2} e^{3 t} \\ x_{3}(t) = C_{3} e^{- t} \end{gathered}\)

Final answer:

\\(egin{gathered} x_{1}(t) = C_{1} e^{2 t} \\ x_{2}(t) = C_{2} e^{3 t} \\ x_{3}(t) = C_{3} e^{- t} \end{gathered}\\)
See Example 1 Hide Example 1

Problem

\( ec{x}\,' = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{pmatrix} ec{x}\)

Approach

Step-by-step method

  1. Write the system in matrix form x' = A x.
  2. Use the eigenvalue method: x(t) is a combination of e^(λ t) v.
  3. Find the eigenvalues of A.
  4. Find an eigenvector for each eigenvalue.
  5. Combine them into the general solution.

Step 1

Step 1 - Write the system in matrix form x' = A x.

In this problem: The system in matrix form is \( ec{x}\,' = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{pmatrix} ec{x}\), with \(A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{pmatrix}\).

\( ec{x}\,' = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 1 & 3 \end{pmatrix} ec{x}\)

Step 2

Step 2 - Use the eigenvalue method: x(t) is a combination of e^(λ t) v.

In this problem: Each eigenvalue and eigenvector pair contributes one term to the solution.

\(\vec{x}(t) = C_{1} e^{\lambda_{1} t} \vec{v}_{1} + C_{2} e^{\lambda_{2} t} \vec{v}_{2} + \cdots\)

Step 3

Step 3 - Find the eigenvalues of A.

In this problem: Solving det(A - \(\lambda\) I) = 0 gives \(\lambda = 1,\; \lambda = 2,\; \lambda = 3\).

\(\lambda = 1,\; \lambda = 2,\; \lambda = 3\)

Step 4a

Step 4a - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 1\), an eigenvector is \(\left[\begin{matrix}2\\-2\\1\end{matrix}\right]\).

\(\lambda = 1:\; ec{v} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}\)

Step 4b

Step 4b - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 2\), an eigenvector is \(\left[\begin{matrix}0\\-1\\1\end{matrix}\right]\).

\(\lambda = 2:\; ec{v} = \begin{pmatrix} 0 \\ -1 \\ 1 \end{pmatrix}\)

Step 4c

Step 4c - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 3\), an eigenvector is \(\left[\begin{matrix}0\\0\\1\end{matrix}\right]\).

\(\lambda = 3:\; ec{v} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\)

Step 5

Step 5 - Combine them into the general solution.

In this problem: Combine the eigenvalue terms into the general solution with constants C1, C2, and so on.

\(egin{gathered} x_{1}(t) = 2 C_{1} e^{t} \\ x_{2}(t) = \left(- 2 C_{1} - C_{2} e^{t}\right) e^{t} \\ x_{3}(t) = \left(C_{1} + C_{2} e^{t} + C_{3} e^{2 t}\right) e^{t} \end{gathered}\)

Final Answer

\(egin{gathered} x_{1}(t) = 2 C_{1} e^{t} \\ x_{2}(t) = \left(- 2 C_{1} - C_{2} e^{t}\right) e^{t} \\ x_{3}(t) = \left(C_{1} + C_{2} e^{t} + C_{3} e^{2 t}\right) e^{t} \end{gathered}\)
See Example 2 Hide Example 2

Problem

\( ec{x}\,' = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix} ec{x}\)

Approach

Step-by-step method

  1. Write the system in matrix form x' = A x.
  2. Use the eigenvalue method: x(t) is a combination of e^(λ t) v.
  3. Find the eigenvalues of A.
  4. Find an eigenvector for each eigenvalue.
  5. Combine them into the general solution.

Step 1

Step 1 - Write the system in matrix form x' = A x.

In this problem: The system in matrix form is \( ec{x}\,' = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix} ec{x}\), with \(A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix}\).

\( ec{x}\,' = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{pmatrix} ec{x}\)

Step 2

Step 2 - Use the eigenvalue method: x(t) is a combination of e^(λ t) v.

In this problem: Each eigenvalue and eigenvector pair contributes one term to the solution.

\(\vec{x}(t) = C_{1} e^{\lambda_{1} t} \vec{v}_{1} + C_{2} e^{\lambda_{2} t} \vec{v}_{2} + \cdots\)

Step 3

Step 3 - Find the eigenvalues of A.

In this problem: Solving det(A - \(\lambda\) I) = 0 gives \(\lambda = -1,\; \lambda = 2,\; \lambda = 3\).

\(\lambda = -1,\; \lambda = 2,\; \lambda = 3\)

Step 4a

Step 4a - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = -1\), an eigenvector is \(\left[\begin{matrix}0\\0\\1\end{matrix}\right]\).

\(\lambda = -1:\; ec{v} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\)

Step 4b

Step 4b - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 2\), an eigenvector is \(\left[\begin{matrix}1\\0\\0\end{matrix}\right]\).

\(\lambda = 2:\; ec{v} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\)

Step 4c

Step 4c - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 3\), an eigenvector is \(\left[\begin{matrix}0\\1\\0\end{matrix}\right]\).

\(\lambda = 3:\; ec{v} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\)

Step 5

Step 5 - Combine them into the general solution.

In this problem: Combine the eigenvalue terms into the general solution with constants C1, C2, and so on.

\(egin{gathered} x_{1}(t) = C_{1} e^{2 t} \\ x_{2}(t) = C_{2} e^{3 t} \\ x_{3}(t) = C_{3} e^{- t} \end{gathered}\)

Final Answer

\(egin{gathered} x_{1}(t) = C_{1} e^{2 t} \\ x_{2}(t) = C_{2} e^{3 t} \\ x_{3}(t) = C_{3} e^{- t} \end{gathered}\)
Coefficient matrix A