Internal problem ID [10215]
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 }\left (t \right )-a y^{\prime }\left (t \right )+b x \left (t \right )&=0\\ y^{\prime \prime }\left (t \right )+a x^{\prime }\left (t \right )+b y \left (t \right )&=0 \end {align*}
✓ Solution by Maple
Time used: 0.094 (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],singsol=all)
\begin{align*} x \left (t \right ) &= -\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}}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{1} a^{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 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, 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}}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{3} a^{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 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{4} a^{2}+4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{1} b -4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}-2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{2} b +4 \,{\mathrm e}^{-\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{3} b -4 \,{\mathrm e}^{\frac {\sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, t}{2}} \sqrt {-2 a^{2}+2 \sqrt {a^{2} \left (a^{2}+4 b \right )}-4 b}\, c_{4} b}{8 a b} \\ y \left (t \right ) &= 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}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.263 (sec). Leaf size: 3522
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]
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