Solve the system of differential equations: dot…
Solve the system of differential equations: $$ \dot x_1=x_1-x_2-x_3,\qquad \dot x_2=x_1+x_2,\qquad \dot x_3=3x_1+x_3 $$
Explanation of Method
The characteristic cubic factors as $(1-\lambda)[(1-\lambda)^2+4]$, giving one real eigenvalue and a complex-conjugate pair; the complex eigenvector is split into real and imaginary parts to build two real oscillatory solutions.
- Write the matrix $A$ and factor the characteristic cubic.
- Find the real eigenvector for $\lambda_1=1$.
- Find the complex eigenvector for $\lambda=1+2i$ and split it into real and imaginary parts.
- Combine into two real oscillatory solutions and assemble the general solution.
Step 1 · Characteristic equation
With $A=\begin{pmatrix}1&-1&-1\\1&1&0\\3&0&1\end{pmatrix}$, expanding the determinant gives
$$ (1-\lambda)\bigl[(1-\lambda)^2+4\bigr]=0, $$
so $\lambda_1=1$ and $\lambda_{2,3}=1\pm2i$.
Step 2 · Real eigenvector
Solving $(A-I)\mathbf v=0$ gives
$$ \mathbf v_1=\begin{pmatrix}0\\-1\\1\end{pmatrix}. $$
Step 3 · Complex eigenvector
Solving $(A-(1+2i)I)\mathbf v=0$ gives $\mathbf v=(2i,1,3)=\mathbf a+i\mathbf b$ with $\mathbf a=(0,1,3)$, $\mathbf b=(2,0,0)$.
Step 4 · Real oscillatory solutions
With $\alpha=1,\ \beta=2$, the two real solutions are $e^t(\mathbf a\cos2t-\mathbf b\sin2t)$ and $e^t(\mathbf a\sin2t+\mathbf b\cos2t)$.
Step 5 · Verification
Differentiating the general solution and substituting into all three equations confirms each holds identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda_1=1,\ \lambda_{2,3}=1\pm2i$ |
| Real eigenvector | $(0,-1,1)$ |
| Complex eigenvector | $(2i,1,3)$ |
Additional Notes
- The real eigenvalue drives pure exponential growth in the direction $(0,-1,1)$, while the complex pair produces a spiral (growing oscillation) in the complementary plane.
- Because $\operatorname{Re}(\lambda_{2,3})=1>0$, every non-trivial trajectory grows without bound as $t\to\infty$.
- Verified by direct substitution of the explicit general solution into all three original equations.
Result
The general solution of the system is: