60.9.13 problem 1868

Internal problem ID [11792]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1868
Date solved : Sunday, March 30, 2025 at 09:15:29 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.185 (sec). Leaf size: 63

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

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

✓ Mathematica. Time used: 0.154 (sec). Leaf size: 166

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

\begin{align*} x(t)\to e^{-4 t} \left ((t+1) \int _1^te^{5 K[1]} \left (-e^{K[1]} (K[1]-1)-K[1]\right )dK[1]+t \int _1^te^{5 K[2]} \left (e^{K[2]} K[2]+K[2]+1\right )dK[2]+c_1 t+c_2 t+c_1\right ) \\ y(t)\to e^{-4 t} \left (-t \int _1^te^{5 K[1]} \left (-e^{K[1]} (K[1]-1)-K[1]\right )dK[1]-(t-1) \int _1^te^{5 K[2]} \left (e^{K[2]} K[2]+K[2]+1\right )dK[2]+c_1 (-t)-c_2 t+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.205 (sec). Leaf size: 63

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

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