60.9.36 problem 1891
Internal
problem
ID
[11890]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1891
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 )+6 x \left (t \right )+7 y \left (t \right )&=0\\ \frac {d^{2}}{d t^{2}}y \left (t \right )+3 x \left (t \right )+2 y \left (t \right )&=2 t \end{align*}
✓ Solution by Maple
Time used: 0.085 (sec). Leaf size: 63
dsolve([diff(x(t),t,t)+6*x(t)+7*y(t)=0,diff(y(t),t,t)+3*x(t)+2*y(t)=2*t],singsol=all)
\begin{align*}
x \left (t \right ) &= \frac {14 t}{9}+c_{1} {\mathrm e}^{t}+\cos \left (3 t \right ) c_{2} +c_3 \,{\mathrm e}^{-t}+c_4 \sin \left (3 t \right ) \\
y \left (t \right ) &= -c_{1} {\mathrm e}^{t}+\frac {3 \cos \left (3 t \right ) c_{2}}{7}-c_3 \,{\mathrm e}^{-t}+\frac {3 c_4 \sin \left (3 t \right )}{7}-\frac {4 t}{3} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.695 (sec). Leaf size: 664
DSolve[{D[x[t],{t,2}]+6*x[t]+7*y[t]==0,D[y[t],{t,2}]+3*x[t]+2*y[t]==2*t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {1}{60} e^{-t} \left (-21 \left (e^{2 t}-2 e^t \cos (3 t)+1\right ) \int _1^t-\frac {1}{10} e^{-K[3]} K[3] \left (2 e^{K[3]} \sin (3 K[3])+7 e^{2 K[3]}-7\right )dK[3]+\left (9 e^{2 t}+14 e^t \sin (3 t)-9\right ) \int _1^t-\frac {7}{10} e^{-K[2]} \left (-2 e^{K[2]} \cos (3 K[2])+e^{2 K[2]}+1\right ) K[2]dK[2]+7 \left (-3 e^{2 t}+2 e^t \sin (3 t)+3\right ) \int _1^t\frac {1}{10} e^{-K[4]} \left (6 e^{K[4]} \cos (3 K[4])+7 e^{2 K[4]}+7\right ) K[4]dK[4]+3 \left (3 e^{2 t}+14 e^t \cos (3 t)+3\right ) \int _1^t\frac {7}{30} e^{-K[1]} K[1] \left (-2 e^{K[1]} \sin (3 K[1])+3 e^{2 K[1]}-3\right )dK[1]-21 c_3 \left (e^{2 t}-2 e^t \cos (3 t)+1\right )+3 c_1 \left (3 e^{2 t}+14 e^t \cos (3 t)+3\right )+7 c_4 \left (-3 e^{2 t}+2 e^t \sin (3 t)+3\right )+c_2 \left (9 e^{2 t}+14 e^t \sin (3 t)-9\right )\right ) \\
y(t)\to \frac {1}{20} e^{-t} \left (-3 \left (e^{2 t}-2 e^t \cos (3 t)+1\right ) \int _1^t\frac {7}{30} e^{-K[1]} K[1] \left (-2 e^{K[1]} \sin (3 K[1])+3 e^{2 K[1]}-3\right )dK[1]+\left (-3 e^{2 t}+2 e^t \sin (3 t)+3\right ) \int _1^t-\frac {7}{10} e^{-K[2]} \left (-2 e^{K[2]} \cos (3 K[2])+e^{2 K[2]}+1\right ) K[2]dK[2]+\left (7 e^{2 t}+2 e^t \sin (3 t)-7\right ) \int _1^t\frac {1}{10} e^{-K[4]} \left (6 e^{K[4]} \cos (3 K[4])+7 e^{2 K[4]}+7\right ) K[4]dK[4]+\left (7 e^{2 t}+6 e^t \cos (3 t)+7\right ) \int _1^t-\frac {1}{10} e^{-K[3]} K[3] \left (2 e^{K[3]} \sin (3 K[3])+7 e^{2 K[3]}-7\right )dK[3]-3 c_1 \left (e^{2 t}-2 e^t \cos (3 t)+1\right )+c_3 \left (7 e^{2 t}+6 e^t \cos (3 t)+7\right )+c_2 \left (-3 e^{2 t}+2 e^t \sin (3 t)+3\right )+c_4 \left (7 e^{2 t}+2 e^t \sin (3 t)-7\right )\right ) \\
\end{align*}