Internal problem ID [10188]
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 }\left (t \right )&=a_{1} x \left (t \right )+b_{1} y \left (t \right )+c_{1}\\ y^{\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: 0.078 (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},singsol=all)
\begin{align*} x \left (t \right ) &= {\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 \left (t \right ) &= \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}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.359 (sec). Leaf size: 926
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 {2 e^{-\frac {1}{2} t \left (\sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}+\text {a1}+\text {b2}\right )} \left (2 \text {b2} \text {c1} \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2} e^{\frac {1}{2} t \left (\sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}+\text {a1}+\text {b2}\right )}+\text {b1} \left (-2 \text {c2} \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2} e^{\frac {1}{2} t \left (\sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}+\text {a1}+\text {b2}\right )}+\text {a2} e^{t (\text {a1}+\text {b2})} \left (\text {a1} c_1 \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )+c_1 \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2} \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}+1\right )+2 \text {b1} c_2 \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )\right )\right )+\text {a1} \text {b2}^2 c_1 e^{t (\text {a1}+\text {b2})} \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )-\text {b2} e^{t (\text {a1}+\text {b2})} \left (\text {a1}^2 c_1 \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )+\text {a1} c_1 \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2} \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}+1\right )+2 \text {a1} \text {b1} c_2 \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )+\text {a2} \text {b1} c_1 \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )\right )\right )}{(4 \text {a2} \text {b1}-4 \text {a1} \text {b2}) \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}} \\ y(t)\to \frac {e^{-\frac {1}{2} t \left (\sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}+\text {a1}+\text {b2}\right )} \left (4 \text {a2}^2 \text {b1} c_1 e^{t (\text {a1}+\text {b2})} \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )-4 \text {a2} \text {c1} \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2} e^{\frac {1}{2} t \left (\sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}+\text {a1}+\text {b2}\right )}+4 \text {a1} \text {c2} \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2} e^{\frac {1}{2} t \left (\sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}+\text {a1}+\text {b2}\right )}+2 \text {a2} e^{t (\text {a1}+\text {b2})} \left (\text {b1} c_2 \left (\text {b2} \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )+\sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2} \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}+1\right )\right )-\text {a1} (\text {b1} c_2+2 \text {b2} c_1) \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )\right )-2 \text {a1} \text {b2} c_2 e^{t (\text {a1}+\text {b2})} \left (\text {a1} \left (-e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}\right )+\text {b2} \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}-1\right )+\sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2} \left (e^{t \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}}+1\right )+\text {a1}\right )\right )}{(4 \text {a2} \text {b1}-4 \text {a1} \text {b2}) \sqrt {\text {a1}^2-2 \text {a1} \text {b2}+4 \text {a2} \text {b1}+\text {b2}^2}} \\ \end{align*}