Internal problem ID [9439]
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 }\relax (t )&=a x \relax (t )+b y \relax (t )\\ y^{\prime }\relax (t )&=c x \relax (t )+b y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.089 (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)},{x(t), y(t)}, singsol=all)
\[ x \relax (t ) = c_{1} {\mathrm e}^{\frac {\left (a +b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}} \] \[ y \relax (t ) = \left (\frac {1}{2}+\frac {\frac {\sqrt {a^{2}-2 a b +b^{2}+4 b c}}{2}-\frac {a}{2}}{b}\right ) c_{1} {\mathrm e}^{\frac {\left (a +b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}+\left (\frac {{\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}}{2}+\frac {-\frac {\sqrt {a^{2}-2 a b +b^{2}+4 b c}\, {\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}}}{2}-\frac {{\mathrm e}^{-\frac {\left (-a -b +\sqrt {a^{2}-2 a b +b^{2}+4 b c}\right ) t}{2}} a}{2}}{b}\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 188
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 e^{\frac {1}{2} t (a+b)} \left (c_1 \cosh \left (\frac {1}{2} t \sqrt {(a-b)^2+4 b c}\right )+\frac {(c_1 (a-b)+2 b c_2) \sinh \left (\frac {1}{2} t \sqrt {(a-b)^2+4 b c}\right )}{\sqrt {(a-b)^2+4 b c}}\right ) \\ y(t)\to e^{\frac {1}{2} t (a+b)} \left (c_2 \cosh \left (\frac {1}{2} t \sqrt {(a-b)^2+4 b c}\right )+\frac {(c_2 (b-a)+2 c c_1) \sinh \left (\frac {1}{2} t \sqrt {(a-b)^2+4 b c}\right )}{\sqrt {(a-b)^2+4 b c}}\right ) \\ \end{align*}