Internal problem ID [10246]
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 }\left (t \right )&=x \left (t \right ) y \left (t \right ) a +x \left (t \right ) b\\ y^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right ) c +y \left (t \right ) d \end {align*}
✓ Solution by Maple
Time used: 0.672 (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*} \{y \left (t \right ) = 0\} \{x \left (t \right ) = c_{1} {\mathrm e}^{t b}\} \end{align*} \begin{align*} \left \{y \left (t \right ) = \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} d \left (\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {\textit {\_a} a}{d}} \textit {\_a}^{\frac {b}{d}} {\mathrm e}^{\frac {c_{1}}{d}} {\mathrm e}^{-1}}{d}\right )+1\right )}d \textit {\_a} \right )+t +c_{2} \right )\right \} \left \{x \left (t \right ) = \frac {-d y \left (t \right )+\frac {d}{d t}y \left (t \right )}{c y \left (t \right )}\right \} \end{align*}
✓ Solution by Mathematica
Time used: 0.359 (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 W\left (\frac {a \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\&\right ][b t+c_2]{}^{\frac {d}{b}} \exp \left (\frac {c \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\&\right ][b t+c_2]+c_1}{b}\right )}{b}\right )}{a} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\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*}