Internal problem ID [10633]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-2. Equations containing
arbitrary functions and their derivatives.
Problem number: 40.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-f^{\prime }\left (x \right ) y^{2}-a \,{\mathrm e}^{\lambda x} f \left (x \right ) y=a \,{\mathrm e}^{\lambda x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 114
dsolve(diff(y(x),x)=diff(f(x),x)*y(x)^2+a*exp(lambda*x)*f(x)*y(x)+a*exp(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-f \left (x \right ) {\mathrm e}^{\int \frac {{\mathrm e}^{x \lambda } f \left (x \right )^{2} a -2 \frac {d}{d x}f \left (x \right )}{f \left (x \right )}d x}-\left (\int \left (\frac {d}{d x}f \left (x \right )\right ) {\mathrm e}^{a \left (\int {\mathrm e}^{x \lambda } f \left (x \right )d x \right )-2 \left (\int \frac {\frac {d}{d x}f \left (x \right )}{f \left (x \right )}d x \right )}d x \right )-c_{1}}{f \left (x \right ) \left (c_{1} +\int \left (\frac {d}{d x}f \left (x \right )\right ) {\mathrm e}^{a \left (\int {\mathrm e}^{x \lambda } f \left (x \right )d x \right )-2 \left (\int \frac {\frac {d}{d x}f \left (x \right )}{f \left (x \right )}d x \right )}d x \right )} \]
✓ Solution by Mathematica
Time used: 84.356 (sec). Leaf size: 167
DSolve[y'[x]==f'[x]*y[x]^2+a*Exp[\[Lambda]*x]*f[x]*y[x]+a*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {a \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]\right )}{a f\left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right ) \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]\right )-c_1 \lambda } \]