Solve the system of differential equations: (dx_1)/(dt) =…
Solve the system of differential equations: $$ \frac{dx_1}{dt} = x_1 - 5x_2,\qquad \frac{dx_2}{dt} = 2x_1 - x_2 $$
Explanation of Method
The coefficient matrix has purely imaginary eigenvalues, so the solution is purely oscillatory; a complex eigenvector is split into real and imaginary parts to construct two real, independent solutions.
- Write the system in matrix form and solve the characteristic equation.
- Find a complex eigenvector for one of the eigenvalues.
- Split it into real and imaginary parts to build real solutions.
- Combine into the general (oscillatory) solution.
Step 1 · Characteristic equation
With $A=\begin{pmatrix}1&-5\\2&-1\end{pmatrix}$,
$$ \det(A-\lambda I)=\lambda^2+9=0 \implies \lambda=\pm3i. $$
Step 2 · Complex eigenvector
For $\lambda=3i$, solving $(A-3iI)\mathbf v=0$ gives $\mathbf v=(5,\,1-3i)=\mathbf a+i\mathbf b$ with $\mathbf a=(5,1)$, $\mathbf b=(0,-3)$.
Step 3 · Real solutions and general form
With $\alpha=0,\ \beta=3$, the two real solutions are $e^{\alpha t}(\mathbf a\cos\beta t-\mathbf b\sin\beta t)$ and $e^{\alpha t}(\mathbf a\sin\beta t+\mathbf b\cos\beta t)$, giving
$$ x_1(t)=5C_1\cos3t+5C_2\sin3t,\qquad x_2(t)=C_1(\cos3t+3\sin3t)+C_2(\sin3t-3\cos3t). $$
Step 4 · Verification
Differentiating and substituting confirms $\dot x_1=x_1-5x_2$ and $\dot x_2=2x_1-x_2$ hold identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda=\pm3i$ |
| Eigenvector (for $3i$) | $(5,\,1-3i)$ |
| Behaviour | purely oscillatory, period $2\pi/3$ |
Additional Notes
- Purely imaginary eigenvalues mean the origin is a centre: trajectories in the $(x_1,x_2)$-plane are closed curves (ellipses), never growing or decaying.
- The choice of which eigenvalue ($3i$ or $-3i$) to expand makes no difference — both give the same real general solution.
- Verified by direct substitution of the trigonometric solution into both original equations.
Result
The general solution of the system is: