77.1.159 problem 186 (page 297)

Internal problem ID [18049]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 186 (page 297)
Date solved : Tuesday, January 28, 2025 at 11:23:28 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 )&=7 \,{\mathrm e}^{t}-27\\ \frac {d}{d t}y \left (t \right )-2 x \left (t \right )+3 y \left (t \right )&=-3 \,{\mathrm e}^{t}+12 \end{align*}

✓ Solution by Maple

Time used: 0.078 (sec). Leaf size: 70

dsolve([diff(x(t),t)+5*x(t)+y(t)=7*exp(t)-27,diff(y(t),t)-2*x(t)+3*y(t)=-3*exp(t)+12],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} \sin \left (t \right ) c_{2} +{\mathrm e}^{-4 t} \cos \left (t \right ) c_{1} -\frac {93}{17}+\frac {31 \,{\mathrm e}^{t}}{26} \\ y \left (t \right ) &= -{\mathrm e}^{-4 t} \sin \left (t \right ) c_{2} -{\mathrm e}^{-4 t} \cos \left (t \right ) c_{2} -{\mathrm e}^{-4 t} \cos \left (t \right ) c_{1} +{\mathrm e}^{-4 t} \sin \left (t \right ) c_{1} -\frac {2 \,{\mathrm e}^{t}}{13}+\frac {6}{17} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.213 (sec). Leaf size: 79

DSolve[{D[x[t],t]+5*x[t]+y[t]==7*Exp[t]-27,D[y[t],t]-2*x[t]+3*y[t]==3*Exp[t]+12},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {25 e^t}{26}+c_1 e^{-4 t} \cos (t)-(c_1+c_2) e^{-4 t} \sin (t)-\frac {93}{17} \\ y(t)\to \frac {16 e^t}{13}+c_2 e^{-4 t} \cos (t)+(2 c_1+c_2) e^{-4 t} \sin (t)+\frac {6}{17} \\ \end{align*}