Internal problem ID [9687]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and
exponential functions
Problem number: 22.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a x \,{\mathrm e}^{\lambda x} y-{\mathrm e}^{\lambda x} a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 76
dsolve(diff(y(x),x)=y(x)^2+a*x*exp(lambda*x)*y(x)+a*exp(lambda*x),y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\frac {a x \,{\mathrm e}^{\lambda x}}{\lambda }-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda ^{2}}}}{x^{2} \lambda ^{2} \left (c_{1}-\left (\int \frac {{\mathrm e}^{\frac {a x \,{\mathrm e}^{\lambda x}}{\lambda }-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda ^{2}}}}{x^{2} \lambda ^{2}}d x \right )\right )}-\frac {1}{x} \]
✓ Solution by Mathematica
Time used: 1.707 (sec). Leaf size: 75
DSolve[y'[x]==y[x]^2+a*x*Exp[\[Lambda]*x]*y[x]+a*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x+\frac {e^{\frac {a e^{\lambda x} (\lambda x-1)}{\lambda ^2}}}{\int _1^x\frac {e^{\frac {a e^{\lambda K[1]} (\lambda K[1]-1)}{\lambda ^2}}}{K[1]^2}dK[1]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}