77.1.165 problem 192 (page 298)

Internal problem ID [18055]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 192 (page 298)
Date solved : Tuesday, January 28, 2025 at 11:23:31 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+5 x \left (t \right )+y \left (t \right )&={\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )+3 y \left (t \right )-x \left (t \right )&={\mathrm e}^{2 t} \end{align*}

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 64

dsolve([diff(x(t),t)+5*x(t)+y(t)=exp(t),diff(y(t),t)+3*y(t)-x(t)=exp(2*t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-4 t}+t \,{\mathrm e}^{-4 t} c_{1} +\frac {4 \,{\mathrm e}^{t}}{25}-\frac {{\mathrm e}^{2 t}}{36} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{-4 t}-{\mathrm e}^{-4 t} c_{1} -t \,{\mathrm e}^{-4 t} c_{1} +\frac {{\mathrm e}^{t}}{25}+\frac {7 \,{\mathrm e}^{2 t}}{36} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.143 (sec). Leaf size: 76

DSolve[{D[x[t],t]+5*x[t]+y[t]==Exp[t],D[y[t],t]+3*y[t]-x[t]==Exp[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {4 e^t}{25}-\frac {e^{2 t}}{36}-e^{-4 t} (c_1 (t-1)+c_2 t) \\ y(t)\to \frac {e^t}{25}+\frac {7 e^{2 t}}{36}+e^{-4 t} ((c_1+c_2) t+c_2) \\ \end{align*}