Internal problem ID [10207]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1884.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right )}{4}-\frac {y \left (t \right )}{2}-\frac {t}{2}+\frac {\cos \left (t \right )^{2}}{2}-\frac {1}{4} \end {align*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 69
dsolve([diff(x(t),t)-x(t)+2*y(t)=0,diff(x(t),t,t)-2*diff(y(t),t)=2*t-cos(2*t)],singsol=all)
\begin{align*} x \left (t \right ) &= -t^{2}+8 \,{\mathrm e}^{\frac {t}{2}} c_{1} +\frac {\sin \left (2 t \right )}{34}+\frac {2 \cos \left (2 t \right )}{17}-4 t +2 c_{2} -4 \\ y \left (t \right ) &= -\frac {t^{2}}{2}+2 \,{\mathrm e}^{\frac {t}{2}} c_{1} +\frac {9 \sin \left (2 t \right )}{68}+\frac {\cos \left (2 t \right )}{34}-t +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.032 (sec). Leaf size: 116
DSolve[{x'[t]-x[t]+2*y[t]==0,x''[t]-2*y'[t]==2*t-Cos[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -t^2-4 t+\frac {1}{34} \sin (2 t)+\frac {2}{17} \cos (2 t)+8 c_1 e^{t/2}+8 c_2 e^{t/2}-8-c_2 \\ y(t)\to -\frac {t^2}{2}-t+\frac {9}{68} \sin (2 t)+\frac {1}{34} \cos (2 t)+2 c_1 e^{t/2}+2 c_2 e^{t/2}-2-\frac {c_2}{2} \\ \end{align*}