Internal problem ID [10219]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1896.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y^{\prime \prime \prime }\left (t \right )-y^{\prime \prime }\left (t \right )+2 x^{\prime }\left (t \right )-x \left (t \right )&=t\\ x^{\prime \prime }\left (t \right )-2 x^{\prime }\left (t \right )-y^{\prime }\left (t \right )+y \left (t \right )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 75
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 ) &= -2-6 c_{5} {\mathrm e}^{t}-t -\frac {2 c_{2} {\mathrm e}^{-t}}{3}-2 c_{4} {\mathrm e}^{t} t -3 c_{5} {\mathrm e}^{t} t^{2}-c_{3} {\mathrm e}^{t} \\ y \left (t \right ) &= -2+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} t \,{\mathrm e}^{t}+c_{4} {\mathrm e}^{t} t^{2}+c_{5} {\mathrm e}^{t} t^{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.703 (sec). Leaf size: 246
DSolve[{x''[t]-2*x'[t]-y'[t]+y[t]==0,y'''[t]-y''[t]+2*x'[t]-x[t]==t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{8} e^{-t} \left (e^{2 t} \left (-2 c_3 t^2+2 c_5 t^2+c_1 \left (2 t^2-6 t+7\right )+c_2 \left (2 t^2+6 t+1\right )-2 c_3 t+4 c_4 t-2 c_5 t+c_3-2 c_4+c_5\right )-8 e^t (t+2)+c_1-c_2-c_3+2 c_4-c_5\right ) \\ y(t)\to \frac {1}{48} \left (e^t \left (4 c_3 t^3-4 c_5 t^3+6 c_3 t^2-12 c_4 t^2+6 c_5 t^2+c_1 \left (-4 t^3+18 t^2-18 t+9\right )-c_2 \left (4 t^3+18 t^2-18 t+9\right )-30 c_3 t+12 c_4 t+18 c_5 t+39 c_3+18 c_4-9 c_5\right )+9 (-c_1+c_2+c_3-2 c_4+c_5) e^{-t}-96\right ) \\ \end{align*}