60.9.34 problem 1889
Internal
problem
ID
[11888]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1889
Date
solved
:
Tuesday, January 28, 2025 at 06:24:01 PM
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*}
✓ Solution by Maple
Time used: 0.072 (sec). Leaf size: 71
dsolve([diff(x(t),t,t)+x(t)+y(t)=-5,diff(y(t),t,t)-4*x(t)-3*y(t)=-3],singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 0.533 (sec). Leaf size: 538
DSolve[{D[x[t],{t,2}]+x[t]+y[t]==-5,D[y[t],{t,2}]-4*x[t]-3*y[t]==-3},{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*}