Internal problem ID [9471]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1892.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime \prime }\relax (t )-a y^{\prime }\relax (t )+b x \relax (t )&=0\\ y^{\prime \prime }\relax (t )+a x^{\prime }\relax (t )+b y \relax (t )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 868
dsolve({diff(x(t),t,t)-a*diff(y(t),t)+b*x(t)=0,diff(y(t),t,t)+a*diff(x(t),t)+b*y(t)=0},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = c_{1} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{3} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \] \[ y \relax (t ) = \frac {c_{1} \left (-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}-c_{2} \left (-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}} {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+4 \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} c_{1} a^{2}-4 \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} c_{2} a^{2}+c_{3} \left (-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}} {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}-c_{4} \left (-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b \right )^{\frac {3}{2}} {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}}+4 \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} c_{3} a^{2}-4 \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} c_{4} a^{2}+4 \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, {\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} c_{1} b -4 \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, {\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} c_{2} b +4 \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, {\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} c_{3} b -4 \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, {\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} c_{4} b}{8 b a} \]
✓ Solution by Mathematica
Time used: 0.151 (sec). Leaf size: 714
DSolve[{x''[t]-a*y'[t]+b*x[t]==0,y''[t]+a*x'[t]+b*y[t]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {\left (a^2 (-c_1)+c_1 \sqrt {a^4+4 a^2 b}-2 a c_4\right ) \cosh \left (\frac {t \sqrt {-a^2-\sqrt {a^4+4 a^2 b}-2 b}}{\sqrt {2}}\right )+\left (a^2 c_1+c_1 \sqrt {a^4+4 a^2 b}+2 a c_4\right ) \cosh \left (\frac {t \sqrt {-a^2+\sqrt {a^4+4 a^2 b}-2 b}}{\sqrt {2}}\right )+\frac {\sqrt {2} \left (a^2 c_2+c_2 \sqrt {a^4+4 a^2 b}+2 a b c_3\right ) \sinh \left (\frac {t \sqrt {-a^2-\sqrt {a^4+4 a^2 b}-2 b}}{\sqrt {2}}\right )}{\sqrt {-a^2-\sqrt {a^4+4 a^2 b}-2 b}}+\frac {\sqrt {2} \left (a^2 (-c_2)+c_2 \sqrt {a^4+4 a^2 b}-2 a b c_3\right ) \sinh \left (\frac {t \sqrt {-a^2+\sqrt {a^4+4 a^2 b}-2 b}}{\sqrt {2}}\right )}{\sqrt {-a^2+\sqrt {a^4+4 a^2 b}-2 b}}}{2 \sqrt {a^4+4 a^2 b}} \\ y(t)\to \frac {\left (a^2 (-c_3)+c_3 \sqrt {a^4+4 a^2 b}+2 a c_2\right ) \cosh \left (\frac {t \sqrt {-a^2-\sqrt {a^4+4 a^2 b}-2 b}}{\sqrt {2}}\right )+\left (a^2 c_3+c_3 \sqrt {a^4+4 a^2 b}-2 a c_2\right ) \cosh \left (\frac {t \sqrt {-a^2+\sqrt {a^4+4 a^2 b}-2 b}}{\sqrt {2}}\right )+\frac {\sqrt {2} \left (a^2 c_4+c_4 \sqrt {a^4+4 a^2 b}-2 a b c_1\right ) \sinh \left (\frac {t \sqrt {-a^2-\sqrt {a^4+4 a^2 b}-2 b}}{\sqrt {2}}\right )}{\sqrt {-a^2-\sqrt {a^4+4 a^2 b}-2 b}}+\frac {\sqrt {2} \left (a^2 (-c_4)+c_4 \sqrt {a^4+4 a^2 b}+2 a b c_1\right ) \sinh \left (\frac {t \sqrt {-a^2+\sqrt {a^4+4 a^2 b}-2 b}}{\sqrt {2}}\right )}{\sqrt {-a^2+\sqrt {a^4+4 a^2 b}-2 b}}}{2 \sqrt {a^4+4 a^2 b}} \\ \end{align*}