60.9.41 problem 1896
Internal
problem
ID
[11895]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1896
Date
solved
:
Tuesday, January 28, 2025 at 06:24:03 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )-2 \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )+y \left (t \right )&=0\\ \frac {d^{3}}{d t^{3}}y \left (t \right )-\frac {d^{2}}{d t^{2}}y \left (t \right )+2 \frac {d}{d t}x \left (t \right )-x \left (t \right )&=t \end{align*}
✓ Solution by Maple
Time used: 0.072 (sec). Leaf size: 74
dsolve([diff(x(t),t,t)-2*diff(x(t),t)-diff(y(t),t)+y(t)=0,diff(y(t),t,t,t)-diff(y(t),t,t)+2*diff(x(t),t)-x(t)=t],singsol=all)
\begin{align*}
x \left (t \right ) &= -2-t -\frac {2 c_{2} {\mathrm e}^{-t}}{3}-c_3 \,{\mathrm e}^{t}-2 c_4 t \,{\mathrm e}^{t}-3 c_5 \,{\mathrm e}^{t} t^{2}-6 c_5 \,{\mathrm e}^{t} \\
y \left (t \right ) &= -2+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_3 \,{\mathrm e}^{t} t +c_4 \,t^{2} {\mathrm e}^{t}+c_5 \,{\mathrm e}^{t} t^{3} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.652 (sec). Leaf size: 892
DSolve[{D[x[t],{t,2}]-2*D[x[t],t]-D[y[t],t]+y[t]==0,D[ y[t],{t,3}]-D[y[t],{t,2}]+2*D[x[t],t]-x[t]==t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
x(t)\to \frac {1}{8} e^{-t} \left (\left (e^{2 t} \left (2 t^2-6 t+7\right )+1\right ) \int _1^t\frac {1}{8} e^{-K[1]} K[1] \left (2 K[1]^2+2 K[1]-e^{2 K[1]}+1\right )dK[1]+\left (e^{2 t} \left (2 t^2+6 t+1\right )-1\right ) \int _1^t\frac {1}{8} e^{-K[2]} K[2] \left (2 K[2]^2-2 K[2]+e^{2 K[2]}-1\right )dK[2]-\left (e^{2 t} \left (2 t^2+2 t-1\right )+1\right ) \int _1^t\frac {1}{48} e^{-K[3]} K[3] \left (4 K[3]^3+6 K[3]^2-18 K[3]+9 e^{2 K[3]}-9\right )dK[3]+\left (e^{2 t} \left (2 t^2-2 t+1\right )-1\right ) \int _1^t\frac {1}{48} e^{-K[5]} K[5] \left (4 K[5]^3-18 K[5]^2-18 K[5]+9 e^{2 K[5]}+39\right )dK[5]+2 \left (e^{2 t} (2 t-1)+1\right ) \int _1^t-\frac {1}{48} e^{-K[4]} K[4] \left (-4 K[4]^3+6 K[4]^2+30 K[4]+9 e^{2 K[4]}-9\right )dK[4]+c_1 \left (e^{2 t} \left (2 t^2-6 t+7\right )+1\right )+c_2 \left (e^{2 t} \left (2 t^2+6 t+1\right )-1\right )-c_3 \left (e^{2 t} \left (2 t^2+2 t-1\right )+1\right )+c_5 \left (e^{2 t} \left (2 t^2-2 t+1\right )-1\right )+2 c_4 \left (e^{2 t} (2 t-1)+1\right )\right ) \\
y(t)\to \frac {1}{48} e^{-t} \left (-6 \left (e^{2 t} \left (2 t^2-2 t-3\right )+3\right ) \int _1^t-\frac {1}{48} e^{-K[4]} K[4] \left (-4 K[4]^3+6 K[4]^2+30 K[4]+9 e^{2 K[4]}-9\right )dK[4]-\left (e^{2 t} \left (4 t^3-18 t^2+18 t-9\right )+9\right ) \int _1^t\frac {1}{8} e^{-K[1]} K[1] \left (2 K[1]^2+2 K[1]-e^{2 K[1]}+1\right )dK[1]-\left (e^{2 t} \left (4 t^3+18 t^2-18 t+9\right )-9\right ) \int _1^t\frac {1}{8} e^{-K[2]} K[2] \left (2 K[2]^2-2 K[2]+e^{2 K[2]}-1\right )dK[2]+\left (e^{2 t} \left (4 t^3+6 t^2-30 t+39\right )+9\right ) \int _1^t\frac {1}{48} e^{-K[3]} K[3] \left (4 K[3]^3+6 K[3]^2-18 K[3]+9 e^{2 K[3]}-9\right )dK[3]-\left (e^{2 t} \left (4 t^3-6 t^2-18 t+9\right )-9\right ) \int _1^t\frac {1}{48} e^{-K[5]} K[5] \left (4 K[5]^3-18 K[5]^2-18 K[5]+9 e^{2 K[5]}+39\right )dK[5]-6 c_4 \left (e^{2 t} \left (2 t^2-2 t-3\right )+3\right )-\left (c_1 \left (e^{2 t} \left (4 t^3-18 t^2+18 t-9\right )+9\right )\right )-c_2 \left (e^{2 t} \left (4 t^3+18 t^2-18 t+9\right )-9\right )+c_3 \left (e^{2 t} \left (4 t^3+6 t^2-30 t+39\right )+9\right )-c_5 \left (e^{2 t} \left (4 t^3-6 t^2-18 t+9\right )-9\right )\right ) \\
\end{align*}