Internal problem ID [9699]
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: 34.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}-b n \,x^{n -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 50
dsolve(diff(y(x),x)=a*exp(lambda*x)*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-2 b a \,x^{n} {\mathrm e}^{\lambda x}-2 \,{\mathrm e}^{\lambda x} a c \right ) {\mathrm e}^{-\lambda x}}{2 a}+\frac {1}{c_{1}-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} \]
✓ Solution by Mathematica
Time used: 0.957 (sec). Leaf size: 40
DSolve[y'[x]==a*Exp[\[Lambda]*x]*(y[x]-b*x^n-c)^2+b*n*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\lambda }{-a e^{\lambda x}+c_1 \lambda }+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}