Internal problem ID [9462]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1887.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime \prime }\left (t \right )&=a x \left (t \right )+b y \left (t \right )\\ y^{\prime \prime }\left (t \right )&=c x \left (t \right )+d y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 418
dsolve([diff(x(t),t,t)=a*x(t)+b*y(t),diff(y(t),t,t)=c*x(t)+d*y(t)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = \left (-\frac {d}{2 c}+\frac {\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}+\frac {a}{2}}{c}\right ) c_{4} {\mathrm e}^{\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (-\frac {d}{2 c}+\frac {\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}+\frac {a}{2}}{c}\right ) c_{3} {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (-\frac {d}{2 c}+\frac {-\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}+\frac {a}{2}}{c}\right ) c_{2} {\mathrm e}^{\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+\left (-\frac {d}{2 c}+\frac {-\frac {\sqrt {a^{2}-2 a d +4 b c +d^{2}}}{2}+\frac {a}{2}}{c}\right ) c_{1} {\mathrm e}^{-\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}} \] \[ y \left (t \right ) = c_{1} {\mathrm e}^{-\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {-2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{3} {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2 \sqrt {a^{2}-2 a d +4 b c +d^{2}}+2 a +2 d}\, t}{2}} \]
✓ Solution by Mathematica
Time used: 0.198 (sec). Leaf size: 670
DSolve[{x''[t]==a*x[t]+b*y[t],y''[t]==c*x[t]+d*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {\left (c_1 \left (\sqrt {(a-d)^2+4 b c}-a+d\right )-2 b c_3\right ) \cosh \left (\frac {t \sqrt {-\sqrt {(a-d)^2+4 b c}+a+d}}{\sqrt {2}}\right )+\left (c_1 \left (\sqrt {(a-d)^2+4 b c}+a-d\right )+2 b c_3\right ) \cosh \left (\frac {t \sqrt {\sqrt {(a-d)^2+4 b c}+a+d}}{\sqrt {2}}\right )+\frac {\sqrt {2} \left (c_2 \left (\sqrt {(a-d)^2+4 b c}-a+d\right )-2 b c_4\right ) \sinh \left (\frac {t \sqrt {-\sqrt {(a-d)^2+4 b c}+a+d}}{\sqrt {2}}\right )}{\sqrt {-\sqrt {(a-d)^2+4 b c}+a+d}}+\frac {\sqrt {2} \left (c_2 \left (\sqrt {(a-d)^2+4 b c}+a-d\right )+2 b c_4\right ) \sinh \left (\frac {t \sqrt {\sqrt {(a-d)^2+4 b c}+a+d}}{\sqrt {2}}\right )}{\sqrt {\sqrt {(a-d)^2+4 b c}+a+d}}}{2 \sqrt {(a-d)^2+4 b c}} \\ y(t)\to \frac {\left (c_3 \left (\sqrt {(a-d)^2+4 b c}+a-d\right )-2 c c_1\right ) \cosh \left (\frac {t \sqrt {-\sqrt {(a-d)^2+4 b c}+a+d}}{\sqrt {2}}\right )+\left (c_3 \left (\sqrt {(a-d)^2+4 b c}-a+d\right )+2 c c_1\right ) \cosh \left (\frac {t \sqrt {\sqrt {(a-d)^2+4 b c}+a+d}}{\sqrt {2}}\right )+\frac {\sqrt {2} \left (c_4 \left (\sqrt {(a-d)^2+4 b c}+a-d\right )-2 c c_2\right ) \sinh \left (\frac {t \sqrt {-\sqrt {(a-d)^2+4 b c}+a+d}}{\sqrt {2}}\right )}{\sqrt {-\sqrt {(a-d)^2+4 b c}+a+d}}+\frac {\sqrt {2} \left (c_4 \left (\sqrt {(a-d)^2+4 b c}-a+d\right )+2 c c_2\right ) \sinh \left (\frac {t \sqrt {\sqrt {(a-d)^2+4 b c}+a+d}}{\sqrt {2}}\right )}{\sqrt {\sqrt {(a-d)^2+4 b c}+a+d}}}{2 \sqrt {(a-d)^2+4 b c}} \\ \end{align*}