Solve the system of differential equations: dot x=y+z, dot…
Solve the system of differential equations: $$ \dot x=y+z,\qquad \dot y=3x+z,\qquad \dot z=3x+y $$
Explanation of Method
The matrix has three distinct real eigenvalues, so ordinary diagonalisation applies; the solution also reveals a nice structural identity, $y-z$ decays purely as $e^{-t}$, which is a useful independent check.
- Write the system in matrix form and find the characteristic polynomial.
- Solve the cubic for the three eigenvalues.
- Find an eigenvector for each eigenvalue.
- Assemble the general solution and verify the structural identity for $y-z$.
Step 1 · Matrix form and characteristic equation
With $A=\begin{pmatrix}0&1&1\\3&0&1\\3&1&0\end{pmatrix}$, expanding $\det(A-\lambda I)$ gives
$$ -(\lambda+1)(\lambda-3)(\lambda+2)=0. $$
Three distinct real eigenvalues: $\lambda_1=-1,\ \lambda_2=3,\ \lambda_3=-2$.
Step 2 · Eigenvectors
$$ \mathbf v_1=\begin{pmatrix}0\\-1\\1\end{pmatrix}\ (\lambda=-1),\qquad \mathbf v_2=\begin{pmatrix}2\\3\\3\end{pmatrix}\ (\lambda=3),\qquad \mathbf v_3=\begin{pmatrix}-1\\1\\1\end{pmatrix}\ (\lambda=-2). $$
Each is checked 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 $10\neq0$, so the modes are independent and superpose directly.
Step 4 · Structural check and verification
Note $y-z$ satisfies $\dot y-\dot z=(3x+z)-(3x+y)=-(y-z)$, so $y(t)-z(t)=-2C_1e^{-t}$ for all $t$ — matching the explicit solution exactly. Direct substitution confirms all three original equations hold. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda_1=-1,\ \lambda_2=3,\ \lambda_3=-2$ |
| Structural identity | $y-z=-2C_1e^{-t}$ |
| Structure | diagonalisable |
Additional Notes
- The identity $y-z=-2C_1e^{-t}$ shows that if $y(0)=z(0)$, then $y(t)=z(t)$ for all $t$ — a useful consistency check independent of the eigenvector computation.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: