Solve the system of differential equations: dot x=4y+z, dot…
Solve the system of differential equations: $$ \dot x=4y+z,\qquad \dot y=z,\qquad \dot z=4y $$
Explanation of Method
Because $x$ does not appear on the right-hand side of the $y$- or $z$-equations, the $(y,z)$ pair forms a self-contained $2\times2$ subsystem that can be solved first; $x$ then follows by a single quadrature. Equivalently, the full $3\times3$ eigenvalue problem gives the same answer directly.
- Note that $x$ decouples from the $(y,z)$ subsystem.
- Write the full system in matrix form and find the characteristic polynomial.
- Find an eigenvector for each of the three distinct eigenvalues.
- Assemble the general solution and confirm it matches the decoupled-subsystem approach.
Step 1 · Matrix form and characteristic equation
With $A=\begin{pmatrix}0&4&1\\0&0&1\\0&4&0\end{pmatrix}$, column 1 is entirely zero, so $\lambda=0$ is immediately an eigenvalue. Expanding along column 1,
$$ \det(A-\lambda I)=-\lambda(\lambda^2-4)=-\lambda(\lambda-2)(\lambda+2)=0. $$
Three distinct real eigenvalues: $\lambda_1=0,\ \lambda_2=2,\ \lambda_3=-2$.
Step 2 · Eigenvectors
$$ \mathbf v_1=\begin{pmatrix}1\\0\\0\end{pmatrix}\ (\lambda=0),\qquad \mathbf v_2=\begin{pmatrix}3\\1\\2\end{pmatrix}\ (\lambda=2),\qquad \mathbf v_3=\begin{pmatrix}-1\\1\\-2\end{pmatrix}\ (\lambda=-2). $$
Each is verified directly against $A\mathbf v=\lambda\mathbf v$. ✓
Step 3 · Cross-check via the decoupled subsystem
The $(y,z)$ subsystem $\dot y=z,\ \dot z=4y$ has matrix $\begin{pmatrix}0&1\\4&0\end{pmatrix}$ with eigenvalues $\pm2$, matching $\lambda_2,\lambda_3$ above. Once $y,z$ are known, $x=\int(4y+z)\,dt$ reproduces the same $x(t)$ found from the full $3\times3$ eigenproblem. ✓
Step 4 · Verification
Direct substitution into all three original equations confirms each holds identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda_1=0,\ \lambda_2=2,\ \lambda_3=-2$ |
| Decoupled subsystem | $(y,z)$ with matrix $\begin{pmatrix}0&1\\4&0\end{pmatrix}$ |
| $x$ obtained by | quadrature of $4y+z$, matching the eigenvector approach |
Additional Notes
- Recognizing the decoupling of $x$ from $(y,z)$ gives a useful independent check on the $3\times3$ eigenvector calculation.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: