Solve the system of differential equations: dot x=3x-tfrac12…
Solve the system of differential equations: $$ \dot x=3x-\tfrac12 y-3t^2-\tfrac12 t+\tfrac32,\qquad \dot y=2y-2t-1 $$
Explanation of Method
Because the $y$-equation is triangular (independent of $x$), it can be solved on its own with an integrating factor; substituting the result into the $x$-equation then leaves a single linear first-order ODE for $x$, again solved with an integrating factor.
- Solve the $y$-equation independently using an integrating factor.
- Substitute the result into the $x$-equation.
- Solve the resulting linear ODE for $x$ with an integrating factor.
- Verify both equations directly.
Step 1 · Solve for $y$
With $\dot y-2y=-2t-1$ and integrating factor $e^{-2t}$,
$$ y(t)=C_2e^{2t}+t+1. $$
Step 2 · Substitute into the $x$-equation
Substituting $y$ into the first equation gives
$$ \dot x-3x=-\tfrac{C_2}{2}e^{2t}-3t^2-t+1. $$
Step 3 · Solve for $x$
With integrating factor $e^{-3t}$, integrating the exponential and polynomial parts separately gives
$$ x(t)=C_1e^{3t}+\frac{C_2}{2}e^{2t}+t^2+t. $$
Step 4 · Verification
Substituting back into both original equations confirms each holds identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| System structure | triangular: the $y$-equation is independent of $x$ |
| $y$-equation solution | $y=C_2e^{2t}+t+1$ |
| $x$-equation solution | $x=C_1e^{3t}+\tfrac{C_2}{2}e^{2t}+t^2+t$ |
Additional Notes
- Triangular linear systems (where one equation doesn't reference the other variable) can always be solved by simple sequential first-order integration — no eigenvalue analysis of a coupling matrix is needed.
- The polynomial forcing in the $x$-equation is handled by the standard formula $\int e^{at}t^n\,dt$ applied term by term.
- Verified by direct substitution into both original equations.
Result
The general solution of the system is: