Internal problem ID [9444]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1865.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=a_{1} x \relax (t )+b_{1} y \relax (t )+c_{1}\\ y^{\prime }\relax (t )&=a_{2} x \relax (t )+b_{2} y \relax (t )+c_{2} \end {align*}
✓ Solution by Maple
Time used: 0.112 (sec). Leaf size: 334
dsolve({diff(x(t),t)=a__1*x(t)+b__1*y(t)+c__1,diff(y(t),t)=a__2*x(t)+b__2*y(t)+c__2},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = {\mathrm e}^{\left (\frac {a_{1}}{2}+\frac {b_{2}}{2}+\frac {\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}{2}\right ) t} c_{4}+{\mathrm e}^{\left (\frac {a_{1}}{2}+\frac {b_{2}}{2}-\frac {\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}}{2}\right ) t} c_{3}+\frac {b_{1} c_{2}-b_{2} c_{1}}{a_{1} b_{2}-a_{2} b_{1}} \] \[ y \relax (t ) = \frac {-\frac {a_{1} \left ({\mathrm e}^{\frac {\left (a_{1}+b_{2}+\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\right ) t}{2}} c_{4} \left (a_{1} b_{2}-a_{2} b_{1}\right )+{\mathrm e}^{\frac {\left (a_{1}+b_{2}-\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\right ) t}{2}} c_{3} \left (a_{1} b_{2}-a_{2} b_{1}\right )-b_{2} c_{1}+b_{1} c_{2}\right ) \left (2 a_{1} b_{2}-2 a_{2} b_{1}\right )}{a_{1} b_{2}-a_{2} b_{1}}+\frac {\left (a_{1}+b_{2}+\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\right ) {\mathrm e}^{\frac {\left (a_{1}+b_{2}+\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\right ) t}{2}} c_{4} \left (2 a_{1} b_{2}-2 a_{2} b_{1}\right )}{2}+\frac {\left (a_{1}+b_{2}-\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\right ) {\mathrm e}^{\frac {\left (a_{1}+b_{2}-\sqrt {a_{1}^{2}-2 a_{1} b_{2}+4 a_{2} b_{1}+b_{2}^{2}}\right ) t}{2}} c_{3} \left (2 a_{1} b_{2}-2 a_{2} b_{1}\right )}{2}-c_{1} \left (2 a_{1} b_{2}-2 a_{2} b_{1}\right )}{\left (2 a_{1} b_{2}-2 a_{2} b_{1}\right ) b_{1}} \]
✓ Solution by Mathematica
Time used: 0.962 (sec). Leaf size: 252
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]
\begin{align*} x(t)\to \frac {-c_1 e^{\frac {1}{2} t (\text {a1}+\text {b2})} (\text {a2} \text {b1}-\text {a1} \text {b2}) \cosh \left (\frac {1}{2} t \sqrt {(\text {a1}-\text {b2})^2+4 \text {a2} \text {b1}}\right )+\frac {e^{\frac {1}{2} t (\text {a1}+\text {b2})} (\text {a1} \text {b2}-\text {a2} \text {b1}) (c_1 (\text {a1}-\text {b2})+2 \text {b1} c_2) \sinh \left (\frac {1}{2} t \sqrt {(\text {a1}-\text {b2})^2+4 \text {a2} \text {b1}}\right )}{\sqrt {(\text {a1}-\text {b2})^2+4 \text {a2} \text {b1}}}+\text {b1} \text {c2}-\text {b2} \text {c1}}{\text {a1} \text {b2}-\text {a2} \text {b1}} \\ y(t)\to \frac {\text {a2} \text {c1}-\text {a1} \text {c2}}{\text {a1} \text {b2}-\text {a2} \text {b1}}+e^{\frac {1}{2} t (\text {a1}+\text {b2})} \left (c_2 \cosh \left (\frac {1}{2} t \sqrt {(\text {a1}-\text {b2})^2+4 \text {a2} \text {b1}}\right )+\frac {(c_2 (\text {b2}-\text {a1})+2 \text {a2} c_1) \sinh \left (\frac {1}{2} t \sqrt {(\text {a1}-\text {b2})^2+4 \text {a2} \text {b1}}\right )}{\sqrt {(\text {a1}-\text {b2})^2+4 \text {a2} \text {b1}}}\right ) \\ \end{align*}