54.9.34 problem 1889

Internal problem ID [13112]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1889
Date solved : Sunday, October 12, 2025 at 02:28:24 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )+x \left (t \right )+y \left (t \right )&=-5\\ \frac {d^{2}}{d t^{2}}y \left (t \right )-4 x \left (t \right )-3 y \left (t \right )&=-3 \end{align*}

✓ Maple. Time used: 0.134 (sec). Leaf size: 71

ode:=[diff(diff(x(t),t),t)+x(t)+y(t) = -5, diff(diff(y(t),t),t)-4*x(t)-3*y(t) = -3]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 18+c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{t} t +c_4 \,{\mathrm e}^{-t} t \\ y \left (t \right ) &= -2 c_1 \,{\mathrm e}^{t}-2 c_2 \,{\mathrm e}^{-t}-2 c_3 \,{\mathrm e}^{t} t -2 c_3 \,{\mathrm e}^{t}-2 c_4 \,{\mathrm e}^{-t} t +2 c_4 \,{\mathrm e}^{-t}-23 \\ \end{align*}

✓ Mathematica. Time used: 0.322 (sec). Leaf size: 538

ode={D[x[t],{t,2}]+x[t]+y[t]==-5,D[y[t],{t,2}]-4*x[t]-3*y[t]==-3}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{-t} \left (-2 \left (e^{2 t} (t-1)-t-1\right ) \int _1^t\frac {1}{4} e^{-K[1]} \left (e^{2 K[1]} (23-13 K[1])-13 K[1]-23\right )dK[1]-2 \left (e^{2 t} (t-2)+t+2\right ) \int _1^t\frac {1}{4} e^{-K[2]} \left (-13 K[2]+e^{2 K[2]} (13 K[2]-10)-10\right )dK[2]-\left (e^{2 t}-1\right ) t \int _1^t\frac {1}{2} e^{-K[3]} \left (13 K[3]+e^{2 K[3]} (13 K[3]-10)+10\right )dK[3]-\left (e^{2 t} (t-1)+t+1\right ) \int _1^t\frac {1}{2} e^{-K[4]} \left (13 K[4]-e^{2 K[4]} (13 K[4]+3)-3\right )dK[4]-2 c_1 \left (e^{2 t} (t-1)-t-1\right )-2 c_2 \left (e^{2 t} (t-2)+t+2\right )-c_3 \left (e^{2 t}-1\right ) t-c_4 \left (e^{2 t} (t-1)+t+1\right )\right )\\ y(t)&\to \frac {1}{2} e^{-t} \left (2 \left (e^{2 t}-1\right ) t \int _1^t\frac {1}{4} e^{-K[1]} \left (e^{2 K[1]} (23-13 K[1])-13 K[1]-23\right )dK[1]+2 \left (e^{2 t} (t-1)+t+1\right ) \int _1^t\frac {1}{4} e^{-K[2]} \left (-13 K[2]+e^{2 K[2]} (13 K[2]-10)-10\right )dK[2]+\left (-t+e^{2 t} (t+1)+1\right ) \int _1^t\frac {1}{2} e^{-K[3]} \left (13 K[3]+e^{2 K[3]} (13 K[3]-10)+10\right )dK[3]+\left (e^{2 t}+1\right ) t \int _1^t\frac {1}{2} e^{-K[4]} \left (13 K[4]-e^{2 K[4]} (13 K[4]+3)-3\right )dK[4]+2 c_1 \left (e^{2 t}-1\right ) t+2 c_2 \left (e^{2 t} (t-1)+t+1\right )+c_3 \left (-t+e^{2 t} (t+1)+1\right )+c_4 \left (e^{2 t}+1\right ) t\right ) \end{align*}

✓ Sympy. Time used: 0.208 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) + y(t) + Derivative(x(t), (t, 2)) + 5,0),Eq(-4*x(t) - 3*y(t) + Derivative(y(t), (t, 2)) + 3,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} t e^{t}}{2} - \frac {C_{2} t e^{- t}}{2} - \left (C_{1} - \frac {C_{4}}{2}\right ) e^{t} - \left (C_{2} + \frac {C_{3}}{2}\right ) e^{- t} + 18, \ y{\left (t \right )} = - C_{1} t e^{t} + C_{2} t e^{- t} + \left (C_{1} - C_{4}\right ) e^{t} + \left (C_{2} + C_{3}\right ) e^{- t} - 23\right ] \]