Solve the system of differential equations: dot x=3x-y+z…
Solve the system of differential equations: $$ \dot x=3x-y+z,\qquad \dot y=x+y+z,\qquad \dot z=4x-y+4z $$
Explanation of Method
The coefficient matrix has three distinct real eigenvalues, so the general solution is a linear combination of three exponential modes, one per eigenvector, with no generalized eigenvectors needed.
- Write the system as $\dot{\mathbf u}=A\mathbf u$ and find $\det(A-\lambda I)$.
- Solve the cubic for the three eigenvalues.
- Find an eigenvector for each eigenvalue.
- Assemble and verify the general solution.
Step 1 · Matrix form and characteristic equation
With $A=\begin{pmatrix}3&-1&1\\1&1&1\\4&-1&4\end{pmatrix}$, expanding $\det(A-\lambda I)$ gives
$$ -\lambda^3+8\lambda^2-17\lambda+10=-(\lambda-1)(\lambda-2)(\lambda-5)=0. $$
Three distinct real eigenvalues: $\lambda_1=1,\ \lambda_2=2,\ \lambda_3=5$.
Step 2 · Eigenvectors
Solving $(A-\lambda I)\mathbf v=0$ for each root:
$$ \mathbf v_1=\begin{pmatrix}-1\\-1\\1\end{pmatrix}\ (\lambda=1),\qquad \mathbf v_2=\begin{pmatrix}-1\\2\\3\end{pmatrix}\ (\lambda=2),\qquad \mathbf v_3=\begin{pmatrix}1\\1\\3\end{pmatrix}\ (\lambda=5). $$
Each is checked directly: $A\mathbf v_1=\mathbf v_1$, $A\mathbf v_2=2\mathbf v_2$, $A\mathbf v_3=5\mathbf v_3$. ✓
Step 3 · Linear independence and general solution
The determinant of $[\mathbf v_1\ \mathbf v_2\ \mathbf v_3]$ is $-12\neq0$, so the three modes are independent and the general solution follows directly by superposition.
Step 4 · Verification
Substituting the general solution into all three original equations and collecting the coefficients of $e^{t}$, $e^{2t}$, $e^{5t}$ confirms each equation holds identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda_1=1,\ \lambda_2=2,\ \lambda_3=5$ (distinct) |
| Eigenvectors | $(-1,-1,1),\ (-1,2,3),\ (1,1,3)$ |
| Structure | diagonalisable, no Jordan blocks |
Additional Notes
- Because all three eigenvalues are real and distinct, the matrix is automatically diagonalisable and no generalized eigenvectors are needed.
- Verified by substituting the full solution back into all three original equations.
Result
The general solution of the system is: