Solve the system of differential equations: dot x=y-z-t+3…
Solve the system of differential equations: $$ \dot x=y-z-t+3,\qquad \dot y=7-2z,\qquad \dot z=2y-2t $$
Explanation of Method
Since $x$ does not appear in the $y$- or $z$-equations, the $(y,z)$ pair forms a self-contained linear, non-homogeneous, non-autonomous $2\times2$ subsystem. It is solved first (an oscillator with purely imaginary eigenvalues $\pm2i$, forced by a constant plus a linear-in-$t$ term); $x$ is then recovered by a single quadrature.
- Observe that $x$ decouples: solve the $(y,z)$ subsystem first.
- Find the homogeneous (oscillatory) solution of the $2\times2$ subsystem.
- Find a particular solution of the subsystem using a linear-in-$t$ ansatz.
- Recover $x(t)$ by direct integration of $\dot x=y-z-t+3$.
Step 1 · The decoupled $(y,z)$ subsystem
Write $\dot{\mathbf v}=B\mathbf v+\mathbf g(t)$ with $\mathbf v=(y,z)$, $B=\begin{pmatrix}0&-2\\2&0\end{pmatrix}$, $\mathbf g(t)=(7,-2t)$. The eigenvalues of $B$ are $\mu=\pm2i$, giving purely oscillatory homogeneous solutions
$$ \mathbf v^{(1)}=\begin{pmatrix}\cos2t\\\sin2t\end{pmatrix},\qquad \mathbf v^{(2)}=\begin{pmatrix}-\sin2t\\\cos2t\end{pmatrix}. $$
Step 2 · Particular solution of the subsystem
Since $\mu=0$ is not an eigenvalue of $B$, try $\mathbf v_p=\mathbf p+\mathbf q\,t$. Matching the $t^1$ and $t^0$ coefficients gives $B\mathbf q=(0,2)$ and $B\mathbf p=\mathbf q-(7,0)$, solved by $\mathbf q=(1,0)$ and $\mathbf p=(0,3)$:
$$ \mathbf v_p=\begin{pmatrix}t\\3\end{pmatrix}. $$
Step 3 · General solution for $y,z$
$$ y(t)=C_1\cos2t-C_2\sin2t+t,\qquad z(t)=C_1\sin2t+C_2\cos2t+3. $$
Both are checked directly against $\dot y=-2z+7$ and $\dot z=2y-2t$. ✓
Step 4 · Recover $x$ by quadrature
Substituting into $\dot x=y-z-t+3$, the constant and linear-in-$t$ pieces cancel exactly, leaving
$$ \dot x=(C_1-C_2)\cos2t-(C_1+C_2)\sin2t, $$
which integrates to $x(t)=\tfrac{C_1-C_2}{2}\sin2t+\tfrac{C_1+C_2}{2}\cos2t+C_3$.
Step 5 · Verification
Direct substitution of all three functions into all three original equations confirms every equation holds identically for arbitrary $C_1,C_2,C_3$. ✓
Summary Table
| Quantity | Value |
|---|---|
| Subsystem eigenvalues | $\mu=\pm2i$ (purely oscillatory) |
| Subsystem particular solution | $y_p=t,\ z_p=3$ |
| $x$ obtained by | direct quadrature of $y-z-t+3$ |
Additional Notes
- Recognizing the decoupling of $x$ from $(y,z)$ avoids setting up a full $3\times3$ non-homogeneous system.
- The linear-in-$t$ forcing terms in $\dot x$ cancel exactly once $y_p,z_p$ are substituted, leaving a purely oscillatory integrand for $x$.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: