Internal problem ID [9664]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 77.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }-\alpha \,x^{k} y^{2}-\beta \,x^{s} y+\alpha \,\lambda ^{2} x^{k}-\beta \lambda \,x^{s}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 225
dsolve((a*x^n+b*x^m+c)*diff(y(x),x)=alpha*x^k*y(x)^2+beta*x^s*y(x)-alpha*lambda^2*x^k+beta*lambda*x^s,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\int \frac {\alpha \,x^{k} {\mathrm e}^{\int -\frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{a \,x^{n}+b \,x^{m}+c}d x \right ) {\mathrm e}^{\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x} \lambda +c_{1} {\mathrm e}^{\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x} \lambda +1\right ) {\mathrm e}^{\int -\frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{c_{1}+\int \frac {\alpha \,x^{k} {\mathrm e}^{\int -\frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+b \,x^{m}+c}d x}}{a \,x^{n}+b \,x^{m}+c}d x} \]
✓ Solution by Mathematica
Time used: 7.543 (sec). Leaf size: 389
DSolve[(a*x^n+b*x^m+c)*y'[x]==\[Alpha]*x^k*y[x]^2+\[Beta]*x^s*y[x]-\[Alpha]*\[Lambda]^2*x^k+\[Beta]*\[Lambda]*x^s,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[6]}-\frac {\beta K[5]^s-2 \alpha \lambda K[5]^k}{b K[5]^m+a K[5]^n+c}dK[5]\right ) \left (-\alpha \lambda K[6]^k+\alpha y(x) K[6]^k+\beta K[6]^s\right )}{(k-s) \alpha \beta \left (b K[6]^m+a K[6]^n+c\right ) (\lambda +y(x))}dK[6]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[6]}-\frac {\beta K[5]^s-2 \alpha \lambda K[5]^k}{b K[5]^m+a K[5]^n+c}dK[5]\right ) K[6]^k}{(k-s) \beta \left (b K[6]^m+a K[6]^n+c\right ) (\lambda +K[7])}-\frac {\exp \left (-\int _1^{K[6]}-\frac {\beta K[5]^s-2 \alpha \lambda K[5]^k}{b K[5]^m+a K[5]^n+c}dK[5]\right ) \left (-\alpha \lambda K[6]^k+\alpha K[7] K[6]^k+\beta K[6]^s\right )}{(k-s) \alpha \beta \left (b K[6]^m+a K[6]^n+c\right ) (\lambda +K[7])^2}\right )dK[6]-\frac {\exp \left (-\int _1^x-\frac {\beta K[5]^s-2 \alpha \lambda K[5]^k}{b K[5]^m+a K[5]^n+c}dK[5]\right )}{(k-s) \alpha \beta (\lambda +K[7])^2}\right )dK[7]=c_1,y(x)\right ] \]