Internal problem ID [10452]
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]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}=b n \,x^{-1+n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 54
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)
\[ y \left (x \right ) = \frac {\left (b a \,{\mathrm e}^{\lambda x} x^{n}+{\mathrm e}^{\lambda x} a c -\lambda +\frac {{\mathrm e}^{-\lambda x}}{c_{1} +\frac {{\mathrm e}^{-\lambda x}}{\lambda }}\right ) {\mathrm e}^{-\lambda x}}{a} \]
✓ Solution by Mathematica
Time used: 1.563 (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*}