Internal problem ID [9493]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 9, system of higher order odes
Problem number: 1914.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=\left (a y \relax (t )+b \right ) x \relax (t )\\ y^{\prime }\relax (t )&=\left (c x \relax (t )+d \right ) y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 92
dsolve({diff(x(t),t)=(a*y(t)+b)*x(t),diff(y(t),t)=(c*x(t)+d)*y(t)},{x(t), y(t)}, singsol=all)
\begin{align*} \{x \relax (t ) = 0\} \\ \{y \relax (t ) = c_{1} {\mathrm e}^{d t}\} \\ \end{align*} \begin{align*} \left \{x \relax (t ) = \RootOf \left (-\left (\int _{}^{\textit {\_Z}}\frac {1}{b \textit {\_a} \left (\LambertW \left (\frac {{\mathrm e}^{\frac {c \textit {\_a}}{b}} \textit {\_a}^{\frac {d}{b}} {\mathrm e}^{\frac {c_{1}}{b}} {\mathrm e}^{-1}}{b}\right )+1\right )}d \textit {\_a} \right )+t +c_{2}\right )\right \} \\ \left \{y \relax (t ) = \frac {-b x \relax (t )+\frac {d}{d t}x \relax (t )}{x \relax (t ) a}\right \} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.233 (sec). Leaf size: 201
DSolve[{x'[t]==(a*y[t]+b)*x[t],y'[t]==(c*x[t]+d)*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {b \text {ProductLog}\left (\frac {a \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (\text {ProductLog}\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\&\right ][b t+c_2]{}^{\frac {d}{b}} \exp \left (\frac {c \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (\text {ProductLog}\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\&\right ][b t+c_2]+c_1}{b}\right )}{b}\right )}{a} \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (\text {ProductLog}\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\&\right ][b t+c_2] \\ \end{align*}