Internal problem ID [9477]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1898.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime \prime }\relax (t )+y^{\prime \prime }\relax (t )-x \relax (t )&=0\\ x^{\prime \prime }\relax (t )-x^{\prime }\relax (t )+y^{\prime }\relax (t )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.067 (sec). Leaf size: 73
dsolve({diff(x(t),t,t)-diff(x(t),t)+diff(y(t),t)=0,diff(x(t),t,t)+diff(y(t),t,t)-x(t)=0},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = \left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) c_{3} {\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) c_{4} {\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}}+c_{1} {\mathrm e}^{t} \] \[ y \relax (t ) = c_{2}+c_{3} {\mathrm e}^{\frac {\left (\sqrt {5}+1\right ) t}{2}}+c_{4} {\mathrm e}^{-\frac {\left (\sqrt {5}-1\right ) t}{2}} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 239
DSolve[{x''[t]-x'[t]+y'[t]==0,x''[t]+y''[t]-x[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{10} e^{-\frac {1}{2} \left (\sqrt {5}-1\right ) t} \left (2 c_1 \left (5 e^{\frac {1}{2} \left (1+\sqrt {5}\right ) t}-\sqrt {5} \left (e^{\sqrt {5} t}-1\right )\right )+2 \sqrt {5} c_2 \left (e^{\sqrt {5} t}-1\right )-c_4 \left (\left (5+\sqrt {5}\right ) e^{\sqrt {5} t}-10 e^{\frac {1}{2} \left (1+\sqrt {5}\right ) t}+5-\sqrt {5}\right )\right ) \\ y(t)\to \frac {1}{10} \left (\left (5+\sqrt {5}\right ) c_1-\left (5+\sqrt {5}\right ) c_2-2 \sqrt {5} c_4\right ) e^{-\frac {1}{2} \left (\sqrt {5}-1\right ) t}+\frac {1}{10} \left (-\left (\left (\sqrt {5}-5\right ) c_1\right )+\left (\sqrt {5}-5\right ) c_2+2 \sqrt {5} c_4\right ) e^{\frac {1}{2} \left (1+\sqrt {5}\right ) t}-c_1+c_2+c_3 \\ \end{align*}