Solve the system of differential equations: dot x=-2x-2y-4z…
Solve the system of differential equations: $$ \dot x=-2x-2y-4z,\qquad \dot y=-2x+y-2z,\qquad \dot z=5x+2y+7z $$
Explanation of Method
The characteristic cubic factors cleanly into the three consecutive integers $1,2,3$; with three distinct eigenvalues the matrix is automatically diagonalisable and the solution is a direct superposition.
- Write the system in matrix form and compute the characteristic polynomial.
- 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 $A=\begin{pmatrix}-2&-2&-4\\-2&1&-2\\5&2&7\end{pmatrix}$, expanding $\det(A-\lambda I)$ gives
$$ -\lambda^3+6\lambda^2-11\lambda+6=-(\lambda-1)(\lambda-2)(\lambda-3)=0. $$
Three distinct real eigenvalues: $\lambda_1=1,\ \lambda_2=2,\ \lambda_3=3$.
Step 2 · Eigenvectors
$$ \mathbf v_1=\begin{pmatrix}-2\\-1\\2\end{pmatrix}\ (\lambda=1),\qquad \mathbf v_2=\begin{pmatrix}-1\\0\\1\end{pmatrix}\ (\lambda=2),\qquad \mathbf v_3=\begin{pmatrix}2\\1\\-3\end{pmatrix}\ (\lambda=3). $$
Each is verified directly against $A\mathbf v=\lambda\mathbf v$. ✓
Step 3 · Linear independence and general solution
The determinant of $[\mathbf v_1\ \mathbf v_2\ \mathbf v_3]$ is $1\neq0$, so the three modes superpose directly.
Step 4 · Verification
Substituting the general solution into all three equations and matching coefficients of $e^t,\ e^{2t},\ e^{3t}$ confirms every equation holds. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda_1=1,\ \lambda_2=2,\ \lambda_3=3$ |
| Eigenvectors | $(-2,-1,2),\ (-1,0,1),\ (2,1,-3)$ |
| Structure | diagonalisable, no Jordan blocks |
Additional Notes
- All three modes grow without bound as $t\to\infty$; the $e^{3t}$ term eventually dominates unless $C_3=0$.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: