Internal problem ID [9463]
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 }\relax (t )&=x \relax (t )-2 y \relax (t )\\ y^{\prime }\relax (t )&=\frac {x \relax (t )}{4}-\frac {y \relax (t )}{2}-\frac {t}{2}+\frac {\cos \left (2 t \right )}{4} \end {align*}
✓ Solution by Maple
Time used: 0.115 (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)},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = 2 c_{1} {\mathrm e}^{\frac {t}{2}}-t^{2}+\frac {\sin \left (2 t \right )}{34}+\frac {2 \cos \left (2 t \right )}{17}-4 t +c_{2} \] \[ y \relax (t ) = \frac {c_{1} {\mathrm e}^{\frac {t}{2}}}{2}-t +\frac {\cos \left (2 t \right )}{34}+\frac {9 \sin \left (2 t \right )}{68}+2-\frac {t^{2}}{2}+\frac {c_{2}}{2} \]
✓ Solution by Mathematica
Time used: 0.265 (sec). Leaf size: 93
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 (t+4)+\frac {1}{34} \sin (2 t)+\frac {2}{17} \cos (2 t)+8 (c_1+c_2) e^{t/2}-8-c_2 \\ y(t)\to 2 (c_1+c_2) e^{t/2}+\frac {1}{68} (9 \sin (2 t)+2 \cos (2 t)-34 (t (t+2)+4+c_2)) \\ \end{align*}