Solve the initial value problem, in which the derivatives…
Solve the initial value problem, in which the derivatives appear mixed together rather than isolated: $$ \dot x+2\dot y=17x+8y,\qquad 13\dot x-\dot y=53x+2y,\qquad x(0)=2,\ y(0)=1 $$
Explanation of Method
Because $\dot x$ and $\dot y$ are mixed together on the left-hand side, the system must first be inverted into the standard form $\dot{\mathbf u}=A\mathbf u$ using a $2\times2$ matrix solve; this particular matrix happens to have irrational eigenvalues, so the solution is built with $\cosh$ and $\sinh$ of $\sqrt{105}\,t/6$ rather than simple exponentials of an integer rate.
- Write the mixed-derivative system as $B\dot{\mathbf u}=C\mathbf u$ and solve for $\dot{\mathbf u}=B^{-1}C\,\mathbf u=A\mathbf u$.
- Find the characteristic equation of $A$ and confirm the eigenvalues are irrational.
- Find the eigenvector formula and express the general solution using $\cosh,\sinh$ of the irrational rate.
- Apply the initial conditions to fix the two constants.
- Verify directly against the original (unresolved) mixed-derivative equations.
Step 1 · Resolve for $\dot x,\dot y$
With $B=\begin{pmatrix}1&2\\13&-1\end{pmatrix}$, $C=\begin{pmatrix}17&8\\53&2\end{pmatrix}$, and $\det B=-27$,
$$ A=B^{-1}C=\frac{1}{9}\begin{pmatrix}41&4\\56&34\end{pmatrix}. $$
Step 2 · Characteristic equation
$\det(A-\lambda I)=\lambda^2-\tfrac{25}{3}\lambda+\tfrac{130}{9}=0$, i.e.\ $9\lambda^2-75\lambda+130=0$, with discriminant $945=9\cdot105$, giving the genuinely irrational eigenvalues
$$ \lambda_{1,2}=\frac{25\pm\sqrt{105}}{6}. $$
Step 3 · Eigenvector formula and general solution
For any $\lambda$ satisfying the characteristic equation, $\mathbf v(\lambda)=(4,\,9\lambda-41)$ is a valid eigenvector (verified directly against both rows of $A-\lambda I$). Splitting the two exponential modes into $\cosh,\sinh$ of $\theta=\tfrac{\sqrt{105}}{6}t$ about the common rate $\tfrac{25}{6}$ gives the general solution
$$ x(t)=e^{25t/6}\!\left[K_1\cosh\theta+K_2\sinh\theta\right],\qquad y(t)=e^{25t/6}\!\left[L_1\cosh\theta+L_2\sinh\theta\right], $$
for constants $K_1,K_2,L_1,L_2$ determined by the eigenvector structure and the initial data.
Step 4 · Apply the initial conditions
With $x(0)=2,\ y(0)=1$, solving the resulting linear system for the two free constants gives
$$ x(t)=e^{25t/6}\!\left[2\cosh\theta+\frac{22\sqrt{105}}{315}\sinh\theta\right],\qquad y(t)=e^{25t/6}\!\left[\cosh\theta+\frac{31\sqrt{105}}{45}\sinh\theta\right],\quad \theta=\frac{\sqrt{105}}{6}t. $$
Step 5 · Verification
At $t=0$: $x(0)=2$, $y(0)=1$ ✓. Differentiating and checking directly against the two original, unresolved equations gives $\dot x(0)+2\dot y(0)=\tfrac{86}{9}+\tfrac{292}{9}=42=17(2)+8(1)$ ✓, and $13\dot x(0)-\dot y(0)=\tfrac{1118}{9}-\tfrac{146}{9}=108=53(2)+2(1)$ ✓.
Summary Table
| Quantity | Value |
|---|---|
| Resolved matrix | $A=\tfrac19\begin{pmatrix}41&4\\56&34\end{pmatrix}$ |
| Eigenvalues | $\lambda_{1,2}=\dfrac{25\pm\sqrt{105}}{6}$ (irrational, but exact) |
| Constants from ICs | solved exactly; involve $\sqrt{105}$ in their coefficients |
Additional Notes
- This problem is unusual in that the derivatives are mixed on the left-hand side ($\dot x+2\dot y$, $13\dot x-\dot y$) rather than isolated — the very first step must be a $2\times2$ matrix inversion just to reach standard form.
- The irrational eigenvalues are a genuine feature of the matrix $A=\tfrac19\begin{pmatrix}41&4\\56&34\end{pmatrix}$, not a transcription artefact: the eigenvector formula $\mathbf v(\lambda)=(4,9\lambda-41)$ was verified to satisfy both rows of $(A-\lambda I)\mathbf v=0$ for any root of the characteristic equation, confirming the derivation is exact.
- The final answer was verified directly against the two original mixed-derivative equations at $t=0$ (rather than against the resolved standard-form system), giving an independent check that the matrix inversion in Step 1 was carried out correctly.
Result
The particular solution satisfying the initial data is: