Internal problem ID [9461]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1882.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime \prime }\relax (t )+y^{\prime }\relax (t )-2 y \relax (t )&={\mathrm e}^{2 t}\\ 2 x^{\prime }\relax (t )+y^{\prime }\relax (t )-3 x \relax (t )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 118
dsolve({2*diff(x(t),t)+diff(y(t),t)-3*x(t)=0,diff(x(t),t$2)+diff(y(t),t)-2*y(t)=exp(2*t)},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = \frac {{\mathrm e}^{2 t}}{4}+c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right )+c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) \] \[ y \relax (t ) = -\frac {{\mathrm e}^{2 t}}{8}+c_{1} {\mathrm e}^{t}-\frac {7 c_{2} {\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right )}{4}+\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, t}{2}\right )}{4}-\frac {7 c_{3} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right )}{4}-\frac {c_{3} {\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{2}\right )}{4} \]
✓ Solution by Mathematica
Time used: 1.033 (sec). Leaf size: 199
DSolve[{2*x'[t]+y'[t]-3*x[t]==0,x''[t]+y'[t]-2*y[t]==Exp[2*t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{276} e^{t/2} \left (23 e^{t/2} \left (3 e^t+6 c_1+2 c_2+4 c_3\right )+46 (3 c_1-c_2-2 c_3) \cos \left (\frac {\sqrt {23} t}{2}\right )-2 \sqrt {23} (9 c_1-11 c_2+2 c_3) \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \\ y(t)\to -\frac {1}{552} e^{t/2} \left (23 e^{t/2} \left (3 e^t-4 (3 c_1+c_2+2 c_3)\right )+92 (3 c_1+c_2-4 c_3) \cos \left (\frac {\sqrt {23} t}{2}\right )+4 \sqrt {23} (-33 c_1+25 c_2+8 c_3) \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \\ \end{align*}