Internal problem ID [10211]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1888.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime \prime }\left (t \right )&=a_{1} x \left (t \right )+b_{1} y \left (t \right )+c_{1}\\ y^{\prime \prime }\left (t \right )&=a_{2} x \left (t \right )+b_{2} y \left (t \right )+c_{2} \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 651
dsolve([diff(x(t),t,t)=a__1*x(t)+b__1*y(t)+c__1,diff(y(t),t,t)=a__2*x(t)+b__2*y(t)+c__2],singsol=all)
\begin{align*} x \left (t \right ) &= -\frac {\left (c_{6} a_{1} b_{2}^{2}+\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{6} a_{1} -c_{6} a_{1}^{2}-c_{6} a_{2} b_{1} \right ) b_{2} +\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{6} +c_{6} a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{\frac {\sqrt {2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}+2 a_{1} +2 b_{2}}\, t}{2}}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (c_{5} a_{1} b_{2}^{2}+\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{5} a_{1} -c_{5} a_{1}^{2}-c_{5} a_{2} b_{1} \right ) b_{2} +\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{5} +c_{5} a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}+2 a_{1} +2 b_{2}}\, t}{2}}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (c_{4} a_{1} b_{2}^{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{4} a_{1} -c_{4} a_{1}^{2}-c_{4} a_{2} b_{1} \right ) b_{2} +\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{4} +c_{4} a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (c_{3} a_{1} b_{2}^{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{3} a_{1} -c_{3} a_{1}^{2}-c_{3} a_{2} b_{1} \right ) b_{2} +\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_{3} +c_{3} a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{-\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {-2 a_{2} b_{1} c_{2} +2 a_{2} b_{2} c_{1}}{2 a_{2} \left (a_{1} b_{2} -a_{2} b_{1} \right )} \\ y \left (t \right ) &= \frac {-a_{1} c_{2} +a_{2} c_{1}}{a_{1} b_{2} -a_{2} b_{1}}+c_{3} {\mathrm e}^{-\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}+c_{5} {\mathrm e}^{-\frac {\sqrt {2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}+2 a_{1} +2 b_{2}}\, t}{2}}+c_{6} {\mathrm e}^{\frac {\sqrt {2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}+2 a_{1} +2 b_{2}}\, t}{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 27.36 (sec). Leaf size: 13523
DSolve[{x''[t]==a1*x[t]+b1*y[t]+c1,y''[t]==a2*x[t]+b2*y[t]+c2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
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