Internal problem ID [9467]
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 }\relax (t )&=a_{1} x \relax (t )+b_{1} y \relax (t )+c_{1}\\ y^{\prime \prime }\relax (t )&=a_{2} x \relax (t )+b_{2} y \relax (t )+c_{2} \end {align*}
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 647
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},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = -\frac {-b_{1} c_{2}+b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}}+c_{3} {\mathrm e}^{-\frac {\sqrt {2 a_{1}+2 b_{2}-2 \sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}\, t}{2}}+c_{4} {\mathrm e}^{\frac {\sqrt {2 a_{1}+2 b_{2}-2 \sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}\, t}{2}}+c_{5} {\mathrm e}^{-\frac {\sqrt {2 a_{1}+2 b_{2}+2 \sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}\, t}{2}}+c_{6} {\mathrm e}^{\frac {\sqrt {2 a_{1}+2 b_{2}+2 \sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}\, t}{2}} \] \[ y \relax (t ) = -\frac {\left (-c_{6} a_{1} b_{2}^{2}+\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\, c_{6} a_{1}+c_{6} a_{1}^{2}+c_{6} a_{2} b_{1}\right ) b_{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\, c_{6}-c_{6} a_{1}\right ) a_{2} b_{1}\right ) {\mathrm e}^{\frac {\sqrt {2 a_{1}+2 b_{2}+2 \sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}\, t}{2}}}{2 b_{1} \left (a_{1} b_{2}-a_{2} b_{1}\right )}-\frac {\left (-c_{5} a_{1} b_{2}^{2}+\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\, c_{5} a_{1}+c_{5} a_{1}^{2}+c_{5} a_{2} b_{1}\right ) b_{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\, c_{5}-c_{5} a_{1}\right ) a_{2} b_{1}\right ) {\mathrm e}^{-\frac {\sqrt {2 a_{1}+2 b_{2}+2 \sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}\, t}{2}}}{2 b_{1} \left (a_{1} b_{2}-a_{2} b_{1}\right )}-\frac {\left (-c_{4} a_{1} b_{2}^{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\, c_{4} a_{1}+c_{4} a_{1}^{2}+c_{4} a_{2} b_{1}\right ) b_{2}+\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\, c_{4}-c_{4} a_{1}\right ) a_{2} b_{1}\right ) {\mathrm e}^{\frac {\sqrt {2 a_{1}+2 b_{2}-2 \sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}\, t}{2}}}{2 b_{1} \left (a_{1} b_{2}-a_{2} b_{1}\right )}-\frac {\left (-c_{3} a_{1} b_{2}^{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\, c_{3} a_{1}+c_{3} a_{1}^{2}+c_{3} a_{2} b_{1}\right ) b_{2}+\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\, c_{3}-c_{3} a_{1}\right ) a_{2} b_{1}\right ) {\mathrm e}^{-\frac {\sqrt {2 a_{1}+2 b_{2}-2 \sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}\, t}{2}}}{2 b_{1} \left (a_{1} b_{2}-a_{2} b_{1}\right )}-\frac {2 a_{1} c_{2}-2 a_{2} c_{1}}{2 \left (a_{1} b_{2}-a_{2} b_{1}\right )} \]
✓ Solution by Mathematica
Time used: 21.526 (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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