77.1.165 problem 192 (page 298)

Internal problem ID [17984]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 192 (page 298)
Date solved : Monday, March 31, 2025 at 04:53:09 PM
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*}

✓ Maple. Time used: 0.172 (sec). Leaf size: 64

ode:=[diff(x(t),t)+5*x(t)+y(t) = exp(t), diff(y(t),t)+3*y(t)-x(t) = exp(2*t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} c_2 +{\mathrm e}^{-4 t} t c_1 -\frac {{\mathrm e}^{2 t}}{36}+\frac {4 \,{\mathrm e}^{t}}{25} \\ y \left (t \right ) &= -{\mathrm e}^{-4 t} c_2 -{\mathrm e}^{-4 t} t c_1 -{\mathrm e}^{-4 t} c_1 +\frac {7 \,{\mathrm e}^{2 t}}{36}+\frac {{\mathrm e}^{t}}{25} \\ \end{align*}

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

ode={D[x[t],t]+5*x[t]+y[t]==Exp[t],D[y[t],t]+3*y[t]-x[t]==Exp[2*t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.204 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*x(t) + y(t) - exp(t) + Derivative(x(t), t),0),Eq(-x(t) + 3*y(t) - exp(2*t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{2} t e^{- 4 t} - \left (C_{1} - C_{2}\right ) e^{- 4 t} - \frac {e^{2 t}}{36} + \frac {4 e^{t}}{25}, \ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} t e^{- 4 t} + \frac {7 e^{2 t}}{36} + \frac {e^{t}}{25}\right ] \]