77.1.161 problem 188 (page 297)

Internal problem ID [17972]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 188 (page 297)
Date solved : Thursday, March 13, 2025 at 11:16:46 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*}

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

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = x(t)+exp(t)+exp(-t)]; 
dsolve(ode);
 

\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*}

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

ode={D[x[t],t]==y[t],D[y[t],t]==x[t]+Exp[t]+Exp[-t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.160 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(-x(t) - exp(t) + Derivative(y(t), t) - exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {t e^{t}}{2} - \frac {t e^{- t}}{2} - \left (C_{1} + \frac {1}{4}\right ) e^{- t} + \left (C_{2} - \frac {1}{4}\right ) e^{t}, \ y{\left (t \right )} = \frac {t e^{t}}{2} + \frac {t e^{- t}}{2} + \left (C_{1} - \frac {1}{4}\right ) e^{- t} + \left (C_{2} + \frac {1}{4}\right ) e^{t}\right ] \]