Linear System of ODEs Solver (2x2) Notes

Published on: September 29, 2026

This Linear System ODE Solver Calculator solves a pair of differential equations that feed into each other, and shows the working line by line. Written in matrix form the system asks which directions keep their shape as time passes, and those directions are exactly the eigenvectors of the matrix. Each eigenvalue supplies the exponential that rides along its own eigenvector. Following the same pattern every time keeps each eigenvector paired with the right eigenvalue.

Step-by-step method

  1. Set up the problem.
  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:

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  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 - Set up the problem.

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

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

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

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

Step 4a

Step 4a - Find an eigenvector for each eigenvalue.

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

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

Step 4b

Step 4b - Find an eigenvector for each eigenvalue.

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

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

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.

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

Final Answer

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  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 - Set up the problem.

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

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

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

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

Step 4a

Step 4a - Find an eigenvector for each eigenvalue.

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

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

Step 4b

Step 4b - Find an eigenvector for each eigenvalue.

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

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

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.

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

Final Answer

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  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 - Set up the problem.

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

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

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 \(\lambda = 2,\; \lambda = 4\).

\(\lambda = 2,\; \lambda = 4\)

Step 4a

Step 4a - Find an eigenvector for each eigenvalue.

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

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

Step 4b

Step 4b - Find an eigenvector for each eigenvalue.

In this problem: For \(\lambda = 4\), an eigenvector is \(\left[\begin{matrix}1\\1\end{matrix}\right]\).

\(\lambda = 4:\; ec{v} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\)

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.

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

Final Answer

\(egin{gathered} x_{1}(t) = \left(- C_{1} + C_{2} e^{2 t}\right) e^{2 t} \\ x_{2}(t) = \left(C_{1} + C_{2} e^{2 t}\right) e^{2 t} \end{gathered}\)
See Example 4 Hide Example 4

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  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 - Set up the problem.

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

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

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

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

Step 4a

Step 4a - Find an eigenvector for each eigenvalue.

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

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

Step 4b

Step 4b - Find an eigenvector for each eigenvalue.

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

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

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.

\(egin{gathered} x_{1}(t) = \left(- C_{1} + C_{2} e^{4 t}\right) e^{- t} \\ x_{2}(t) = \left(C_{1} + C_{2} e^{4 t}\right) e^{- t} \end{gathered}\)

Final Answer

\(egin{gathered} x_{1}(t) = \left(- C_{1} + C_{2} e^{4 t}\right) e^{- t} \\ x_{2}(t) = \left(C_{1} + C_{2} e^{4 t}\right) e^{- t} \end{gathered}\)