ODE No. 1914

$$\{x^{\prime }(t)=x(t)(ay(t)+b),y^{\prime }(t)=y(t)(cx(t)+d)\}$$ ✓ Mathematica : cpu = 0.28613 (sec), leaf count = 204

DSolve[{Derivative[1][x][t] == x[t]*(b + a*y[t]), Derivative[1][y][t] == (d + c*x[t])*y[t]},{x[t], y[t]},t]
 

$$\left\{\{y(t)\to \frac{bW\left(\frac{a\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{1}{K[1](W\left(\frac{a{e}^{\frac{c_1}{b}+\frac{cK[1]}{b}}K[1{]}^{\frac{d}{b}}}{b}\right)+1)}dK[1]\mathrm{\&}][bt+c_2]{}^{\frac{d}{b}}\exp (\frac{c\text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{1}{K[1](W\left(\frac{a{e}^{\frac{c_1}{b}+\frac{cK[1]}{b}}K[1{]}^{\frac{d}{b}}}{b}\right)+1)}dK[1]\mathrm{\&}][bt+c_2]}{b}+\frac{c_1}{b})}{b}\right)}{a},x(t)\to \text{InverseFunction}[{\int }_1^{\mathrm{\#}\text{1}}\frac{1}{K[1](W\left(\frac{a{e}^{\frac{c_1}{b}+\frac{cK[1]}{b}}K[1{]}^{\frac{d}{b}}}{b}\right)+1)}dK[1]\mathrm{\&}][bt+c_2]\}\right\}$$ ✓ Maple : cpu = 0.46 (sec), leaf count = 92

dsolve({diff(x(t),t) = (a*y(t)+b)*x(t), diff(y(t),t) = (c*x(t)+d)*y(t)})
 

$$[\{x\left(t\right)=0\},\{y\left(t\right)=c_1{\mathrm{e}}^{dt}\}]$$