60.9.28 problem 1883
Internal
problem
ID
[11882]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1883
Date
solved
:
Tuesday, January 28, 2025 at 06:23:59 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )+x \left (t \right )&=2 t\\ \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {d}{d t}y \left (t \right )-9 x \left (t \right )+3 y \left (t \right )&=\sin \left (2 t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.218 (sec). Leaf size: 79
dsolve([diff(x(t),t)-diff(y(t),t)+x(t)=2*t,diff(x(t),t$2)+diff(y(t),t)-9*x(t)+3*y(t)=sin(2*t)],singsol=all)
\begin{align*}
x \left (t \right ) &= 2 t +4-\frac {2 \cos \left (2 t \right )}{325}-\frac {36 \sin \left (2 t \right )}{325}+c_{1} {\mathrm e}^{t}+{\mathrm e}^{-3 t} c_{2} +c_3 \,{\mathrm e}^{t} t \\
y \left (t \right ) &= \frac {16 \cos \left (2 t \right )}{325}-\frac {37 \sin \left (2 t \right )}{325}+2 c_{1} {\mathrm e}^{t}+\frac {2 \,{\mathrm e}^{-3 t} c_{2}}{3}+2 c_3 \,{\mathrm e}^{t} t -c_3 \,{\mathrm e}^{t}+10+6 t \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.981 (sec). Leaf size: 488
DSolve[{D[x[t],t]-D[y[t],t]+x[t]==2*t,D[x[t],{t,2}]+D[y[t],t]-9*x[t]+3*y[t]==Sin[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {1}{16} e^{-3 t} \left (\left (e^{4 t} (20 t+7)+9\right ) \int _1^t\frac {1}{16} e^{-K[1]} \left (\left (-4 K[1]-3 e^{4 K[1]}+3\right ) \sin (2 K[1])-32 K[1]^2\right )dK[1]+\left (e^{4 t} (4 t+3)-3\right ) \int _1^t\frac {1}{16} e^{-K[2]} \left (\left (-4 K[2]+9 e^{4 K[2]}+7\right ) \sin (2 K[2])-32 (K[2]-1) K[2]\right )dK[2]-3 \left (e^{4 t} (4 t-1)+1\right ) \int _1^t\frac {1}{8} e^{-K[3]} \left (-16 K[3] (2 K[3]+1)-\left (4 K[3]+e^{4 K[3]}-1\right ) \sin (2 K[3])\right )dK[3]+c_1 \left (e^{4 t} (20 t+7)+9\right )+c_2 \left (e^{4 t} (4 t+3)-3\right )-3 c_3 \left (e^{4 t} (4 t-1)+1\right )\right ) \\
y(t)\to \frac {1}{8} e^{-3 t} \left (\left (e^{4 t} (20 t-3)+3\right ) \int _1^t\frac {1}{16} e^{-K[1]} \left (\left (-4 K[1]-3 e^{4 K[1]}+3\right ) \sin (2 K[1])-32 K[1]^2\right )dK[1]+\left (e^{4 t} (4 t+1)-1\right ) \int _1^t\frac {1}{16} e^{-K[2]} \left (\left (-4 K[2]+9 e^{4 K[2]}+7\right ) \sin (2 K[2])-32 (K[2]-1) K[2]\right )dK[2]-\left (3 e^{4 t} (4 t-3)+1\right ) \int _1^t\frac {1}{8} e^{-K[3]} \left (-16 K[3] (2 K[3]+1)-\left (4 K[3]+e^{4 K[3]}-1\right ) \sin (2 K[3])\right )dK[3]+c_1 \left (e^{4 t} (20 t-3)+3\right )+c_2 \left (e^{4 t} (4 t+1)-1\right )-c_3 \left (3 e^{4 t} (4 t-3)+1\right )\right ) \\
\end{align*}