Internal problem ID [10183]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1860.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=a x \left (t \right )+b y \left (t \right )\\ y^{\prime }\left (t \right )&=c x \left (t \right )+b y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 237
dsolve({diff(x(t),t)=a*x(t)+b*y(t),diff(y(t),t)=c*x(t)+b*y(t)},singsol=all)
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (a +b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}} \\ y \left (t \right ) &= \left (\frac {1}{2}+\frac {\frac {\sqrt {a^{2}-2 a b +b^{2}+4 b c}}{2}-\frac {a}{2}}{b}\right ) c_{1} {\mathrm e}^{\frac {\left (a +b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}+\left (\frac {{\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}}{2}+\frac {-\frac {\sqrt {a^{2}-2 a b +b^{2}+4 b c}\, {\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}}{2}-\frac {{\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}} a}{2}}{b}\right ) c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.033 (sec). Leaf size: 362
DSolve[{x'[t]==a*x[t]+b*y[t],y'[t]==c*x[t]+b*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )} \left (a c_1 \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}-1\right )+c_1 \sqrt {a^2-2 a b+b^2+4 b c} \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}+1\right )-b (c_1-2 c_2) \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}-1\right )\right )}{2 \sqrt {a^2-2 a b+b (b+4 c)}} \\ y(t)\to \frac {e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2+4 b c}+a+b\right )} \left (2 c c_1 \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}-1\right )+c_2 \left (a \left (-e^{t \sqrt {a^2-2 a b+b^2+4 b c}}\right )+b \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}-1\right )+\sqrt {a^2-2 a b+b^2+4 b c} \left (e^{t \sqrt {a^2-2 a b+b^2+4 b c}}+1\right )+a\right )\right )}{2 \sqrt {a^2-2 a b+b (b+4 c)}} \\ \end{align*}