Solve the system of differential equations: dot x=2x-y+z…
Solve the system of differential equations: $$ \dot x=2x-y+z,\qquad \dot y=x+z,\qquad \dot z=-3x+y-2z $$
Explanation of Method
The matrix has three distinct real eigenvalues $0,1,-1$, so it is diagonalisable; the $\lambda=0$ mode reflects the fact that $x+y+z$ is a conserved quantity of the system, a useful structural check.
- 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 the general solution and verify the conservation law for $x+y+z$.
Step 1 · Matrix form and characteristic equation
With $A=\begin{pmatrix}2&-1&1\\1&0&1\\-3&1&-2\end{pmatrix}$, expanding $\det(A-\lambda I)$ gives
$$ -\lambda(\lambda-1)(\lambda+1)=0. $$
Three distinct real eigenvalues: $\lambda_1=0,\ \lambda_2=1,\ \lambda_3=-1$.
Step 2 · Eigenvectors
$$ \mathbf v_1=\begin{pmatrix}-1\\-1\\1\end{pmatrix}\ (\lambda=0),\qquad \mathbf v_2=\begin{pmatrix}-1\\0\\1\end{pmatrix}\ (\lambda=1),\qquad \mathbf v_3=\begin{pmatrix}1\\1\\-2\end{pmatrix}\ (\lambda=-1). $$
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 · Conservation law and verification
Since $\dot x+\dot y+\dot z=(2x-y+z)+(x+z)+(-3x+y-2z)=0$, the quantity $x+y+z$ is conserved — consistent with $\lambda_1=0$ contributing $x+y+z=-C_1$, a constant. Direct substitution confirms all three original equations hold. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda_1=0,\ \lambda_2=1,\ \lambda_3=-1$ |
| Conserved quantity | $x+y+z=-C_1$ |
| Structure | diagonalisable |
Additional Notes
- The conservation law $x+y+z=\text{const}$ is an immediate consequence of the zero eigenvalue and gives a fast independent check on the whole solution.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: