Solve the system of differential equations: dot x=3x-2y, dot…
Solve the system of differential equations: $$ \dot x=3x-2y,\qquad \dot y=2x-y+15e^{t}\sqrt{t} $$
Explanation of Method
The homogeneous part has a repeated eigenvalue requiring a generalised eigenvector; combined with the non-elementary forcing $\sqrt t\,e^t$, undetermined coefficients has no simple ansatz, so variation of parameters (with fractional-power integrals) is used instead.
- Find the repeated eigenvalue, its eigenvector, and a generalised eigenvector.
- Build the fundamental matrix from the Jordan-chain solutions and invert it.
- Compute $\Phi^{-1}(t)\mathbf f(t)$, giving powers of $\sqrt t$.
- Integrate using $\int t^{n/2}\,dt$ and assemble the particular solution.
Step 1 · Repeated eigenvalue and Jordan chain
With $A=\begin{pmatrix}3&-2\\2&-1\end{pmatrix}$, $\det(A-\lambda I)=(\lambda-1)^2=0$, a repeated eigenvalue $\lambda=1$ with eigenvector $\mathbf v_1=(1,1)$ and generalised eigenvector $\mathbf v_2=(\tfrac12,0)$:
$$ \Phi(t)=e^t\begin{pmatrix}1&t+\tfrac12\\1&t\end{pmatrix},\qquad \Phi^{-1}(t)=e^{-t}\begin{pmatrix}-2t&2t+1\\2&-2\end{pmatrix}. $$
Step 2 · Compute $\Phi^{-1}\mathbf f$
With $\mathbf f(t)=15e^t\sqrt t\,(0,1)^\top$,
$$ \Phi^{-1}(t)\mathbf f(t)=15\sqrt t\begin{pmatrix}2t+1\\-2\end{pmatrix}. $$
Step 3 · Integrate
Integrating the resulting half-integer powers of $t$,
$$ v_1=12t^{5/2}+10t^{3/2},\qquad v_2=-20t^{3/2}. $$
Step 4 · Particular solution
$$ \mathbf u_p=\Phi(t)\begin{pmatrix}v_1\\v_2\end{pmatrix}=\begin{pmatrix}-8t^{5/2}e^t\\e^t(-8t^{5/2}+10t^{3/2})\end{pmatrix}. $$
Step 5 · Verification
Differentiating using $\dfrac{d}{dt}[t^{n/2}e^t]=e^t(t^{n/2}+\tfrac{n}{2}t^{n/2-1})$ and substituting confirms both original equations hold identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalue | $\lambda=1$ (repeated, needs generalised eigenvector) |
| Forcing | $15e^t\sqrt t$ — not matched by any polynomial/exponential ansatz |
| Particular solution | $-8t^{5/2}e^t$ and $e^t(-8t^{5/2}+10t^{3/2})$ |
Additional Notes
- This combines two separate sources of difficulty — a repeated (defective) eigenvalue and a non-elementary forcing term — both handled simultaneously by variation of parameters once the Jordan-chain fundamental matrix is built.
- The half-integer powers of $t$ in the particular solution trace directly back to the $\sqrt t$ factor in the forcing.
- Verified by direct differentiation (using the general power rule for $t^{n/2}e^t$) and substitution into both original equations.
Result
The general solution of the system is: