Internal problem ID [9867]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary
functions (but not containing their derivatives).
Problem number: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-{\mathrm e}^{\lambda x} f \relax (x ) y^{2}-\left (a f \relax (x )-\lambda \right ) y-b \,{\mathrm e}^{-\lambda x} f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 68
dsolve(diff(y(x),x)=exp(lambda*x)*f(x)*y(x)^2+(a*f(x)-lambda)*y(x)+b*exp(-lambda*x)*f(x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left ({\mathrm e}^{\lambda x} {\mathrm e}^{-\lambda x} a^{2}+\tanh \left (\frac {\sqrt {a^{4}-4 b \,a^{2}}\, \left (a \left (\int f \relax (x )d x \right )+c_{1}\right )}{2 a^{2}}\right ) \sqrt {a^{4}-4 b \,a^{2}}\right ) {\mathrm e}^{-\lambda x}}{2 a} \]
✓ Solution by Mathematica
Time used: 3.71 (sec). Leaf size: 112
DSolve[y'[x]==Exp[\[Lambda]*x]*f[x]*y[x]^2+(a*f[x]-\[Lambda])*y[x]+b*Exp[-\[Lambda]*x]*f[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {\frac {a^2}{b}}+\frac {\sqrt {4 b-a^2} \tan \left (\frac {\sqrt {4 b-a^2} \left (\int _1^x\frac {e^{\lambda K[5]} f(K[5])}{\sqrt {\frac {e^{2 \lambda K[5]}}{b}}}dK[5]+c_1\right )}{2 \sqrt {b}}\right )}{\sqrt {b}}}{2 \sqrt {\frac {e^{2 \lambda x}}{b}}} \\ \end{align*}