Solve the system of differential equations: dot x = y - 5cos…
Solve the system of differential equations: $$ \dot x = y - 5\cos t,\qquad \dot y = 2x + y $$
Explanation of Method
Eliminating $y$ produces a single second-order linear ODE with sinusoidal forcing, solved with the undetermined-coefficients method using $a\sin t+b\cos t$.
- Express $y$ from the first equation and differentiate.
- Substitute into the second equation to obtain one ODE in $x$.
- Solve the homogeneous equation.
- Find a particular solution using a sine/cosine ansatz.
- Recover $y$ from $x$.
Step 1 · Reduce to one equation
From the first equation, $y=\dot x+5\cos t$, so $\dot y=\ddot x-5\sin t$. Substituting into $\dot y=2x+y$ gives
$$ \ddot x-\dot x-2x=5\sin t+5\cos t. $$
Step 2 · Homogeneous solution
The characteristic equation $r^2-r-2=(r-2)(r+1)=0$ gives $r_1=2$, $r_2=-1$, so $x_h=C_1e^{2t}+C_2e^{-t}$.
Step 3 · Particular solution
Trying $x_p=a\sin t+b\cos t$ and matching coefficients of $\sin t$, $\cos t$ gives $a=-2$, $b=-1$:
$$ x_p=-2\sin t-\cos t. $$
Step 4 · General solution for $x$
$$ x(t)=C_1e^{2t}+C_2e^{-t}-2\sin t-\cos t. $$
Step 5 · Recover $y$
Using $y=\dot x+5\cos t$,
$$ y(t)=2C_1e^{2t}-C_2e^{-t}+\sin t+3\cos t. $$
Step 6 · Verification
Substituting into $\dot x=y-5\cos t$ and $\dot y=2x+y$ confirms both equations hold identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Reduced ODE | $\ddot x-\dot x-2x=5\sin t+5\cos t$ |
| Homogeneous roots | $r=2,\,-1$ |
| Particular solution | $-2\sin t-\cos t$ |
Additional Notes
- Since $\pm i$ are not roots of the characteristic equation, the plain $a\sin t+b\cos t$ ansatz applies with no resonance correction.
- The system's forcing was split by matching $\sin t$ and $\cos t$ coefficients separately, giving a linear $2\times2$ system for $a,b$.
- Verified by direct back-substitution into both original equations.
Result
The general solution of the system is: