Solve the system of differential equations: (dx_1)/(dt) =…
Solve the system of differential equations: $$ \frac{dx_1}{dt} = x_1-x_2+x_3,\qquad \frac{dx_2}{dt} = x_1+x_2-x_3,\qquad \frac{dx_3}{dt} = 2x_1-x_2 $$
Explanation of Method
The $3\times3$ coefficient matrix factors into three distinct real eigenvalues, so the general solution is a direct sum of three independent exponential modes, each with its own eigenvector.
- Write the matrix $A$ and expand $\det(A-\lambda I)$.
- Factor the resulting cubic to find the three eigenvalues.
- Find an eigenvector for each eigenvalue.
- Combine into the general solution.
Step 1 · Characteristic equation
With $A=\begin{pmatrix}1&-1&1\\1&1&-1\\2&-1&0\end{pmatrix}$, expanding the determinant gives
$$ -(\lambda^3-2\lambda^2-\lambda+2)=-(\lambda-1)(\lambda-2)(\lambda+1)=0, $$
so $\lambda_1=1,\ \lambda_2=2,\ \lambda_3=-1$.
Step 2 · Eigenvectors
$$ \mathbf v_1=\begin{pmatrix}1\\1\\1\end{pmatrix}\ (\lambda_1=1),\qquad \mathbf v_2=\begin{pmatrix}1\\0\\1\end{pmatrix}\ (\lambda_2=2),\qquad \mathbf v_3=\begin{pmatrix}1\\-3\\-5\end{pmatrix}\ (\lambda_3=-1). $$
Step 3 · General solution
$$ x_1(t)=C_1e^{t}+C_2e^{2t}+C_3e^{-t},\qquad x_2(t)=C_1e^{t}-3C_3e^{-t},\qquad x_3(t)=C_1e^{t}+C_2e^{2t}-5C_3e^{-t}. $$
Step 4 · Verification
Differentiating and substituting confirms all three original equations hold identically for the general solution. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda_1=1,\ \lambda_2=2,\ \lambda_3=-1$ |
| Eigenvectors | $(1,1,1),\ (1,0,1),\ (1,-3,-5)$ |
| Structure | three independent exponential modes |
Additional Notes
- With three distinct real eigenvalues and a full set of independent eigenvectors, no generalised eigenvectors or complex analysis is needed — the most straightforward case for a $3\times3$ linear system.
- The dominant long-term behaviour is governed by the largest eigenvalue $\lambda_2=2$, whose mode grows fastest as $t\to\infty$.
- Verified by direct substitution of the explicit solution into all three original equations.
Result
The general solution of the system is: