Internal problem ID [9475]
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 }\relax (t )-y^{\prime \prime }\relax (t )+2 x^{\prime }\relax (t )-x \relax (t )&=t\\ x^{\prime \prime }\relax (t )-2 x^{\prime }\relax (t )-y^{\prime }\relax (t )+y \relax (t )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.058 (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},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = -2-c_{3} {\mathrm e}^{t}-6 c_{5} {\mathrm e}^{t}-\frac {2 \,{\mathrm e}^{-t} c_{2}}{3}-t -2 c_{4} {\mathrm e}^{t} t -3 c_{5} {\mathrm e}^{t} t^{2} \] \[ y \relax (t ) = -2+c_{1} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{2}+c_{3} t \,{\mathrm e}^{t}+c_{4} {\mathrm e}^{t} t^{2}+c_{5} {\mathrm e}^{t} t^{3} \]
✓ Solution by Mathematica
Time used: 0.434 (sec). Leaf size: 213
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 (-8 e^t (t+2)+e^{2 t} (c_1 (2 (t-3) t+7)+2 t (c_2 (t+3)-c_3 (t+1)+c_5 (t-1)+2 c_4)+c_2+c_3-2 c_4+c_5)+c_1-c_2-c_3+2 c_4-c_5\right ) \\ y(t)\to \frac {1}{48} \left (9 (-c_1+c_2+c_3-2 c_4+c_5) e^{-t}+e^t (c_1 (9-2 (t-3) t (2 t-3))-c_2 (2 t (t (2 t+9)-9)+9)+2 t (c_3 (t (2 t+3)-15)-6 c_4 (t-1)+c_5 ((3-2 t) t+9))+39 c_3+18 c_4-9 c_5)-96\right ) \\ \end{align*}