Internal problem ID [10417]
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]
\[ \boxed {\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }-\alpha \,x^{k} y^{2}-\beta \,x^{s} y=-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s}} \]
✓ 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)
\[ y \left (x \right ) = -\frac {\left (\left (\int \frac {\alpha \,x^{k} {\mathrm e}^{\int -\frac {2 x^{k} \alpha \lambda -\beta \,x^{s}}{a \,x^{n}+b \,x^{m}+c}d x}}{a \,x^{n}+b \,x^{m}+c}d x \right ) {\mathrm e}^{\int \frac {2 x^{k} \alpha \lambda -\beta \,x^{s}}{a \,x^{n}+b \,x^{m}+c}d x} \lambda +c_{1} {\mathrm e}^{\int \frac {2 x^{k} \alpha \lambda -\beta \,x^{s}}{a \,x^{n}+b \,x^{m}+c}d x} \lambda +1\right ) {\mathrm e}^{\int -\frac {2 x^{k} \alpha \lambda -\beta \,x^{s}}{a \,x^{n}+b \,x^{m}+c}d x}}{c_{1} +\int \frac {\alpha \,x^{k} {\mathrm e}^{\int -\frac {2 x^{k} \alpha \lambda -\beta \,x^{s}}{a \,x^{n}+b \,x^{m}+c}d x}}{a \,x^{n}+b \,x^{m}+c}d x} \]
✓ Solution by Mathematica
Time used: 13.649 (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]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha y(x) K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) K[2]^k}{(k-s) \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])}-\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha K[3] K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right )}{(k-s) \alpha \beta (\lambda +K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]