77.1.161 problem 188 (page 297)

Internal problem ID [18051]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 188 (page 297)
Date solved : Tuesday, January 28, 2025 at 11:23:29 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.694 (sec). Leaf size: 41

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

\begin{align*} x \left (t \right ) &= \sinh \left (t \right ) c_{2} +\cosh \left (t \right ) c_{1} +\sinh \left (t \right ) t -\frac {\cosh \left (t \right )}{2} \\ y \left (t \right ) &= \cosh \left (t \right ) c_{2} +\sinh \left (t \right ) c_{1} +\cosh \left (t \right ) t +\frac {\sinh \left (t \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.064 (sec). Leaf size: 106

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

\begin{align*} x(t)\to \frac {1}{4} e^{-t} \left (\left (e^{2 t}-1\right ) \log \left (e^{2 t}\right )+(-1+2 c_1+2 c_2) e^{2 t}-1+2 c_1-2 c_2\right ) \\ y(t)\to \frac {1}{4} e^{-t} \left (\left (e^{2 t}+1\right ) \log \left (e^{2 t}\right )+(1+2 c_1+2 c_2) e^{2 t}-1-2 c_1+2 c_2\right ) \\ \end{align*}