Internal problem ID [9613]
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: 30.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}-b \,x^{m} y-x^{m} b c +a \,c^{2} x^{n}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 149
dsolve(diff(y(x),x)=a*x^n*y(x)^2+b*x^m*y(x)+b*c*x^m-a*c^2*x^n,y(x), singsol=all)
\[ c_{1} +\int _{}^{x}-\frac {\left (-a c \,\textit {\_a}^{1+n}-a y \left (x \right ) \textit {\_a}^{1+n}\right ) {\mathrm e}^{-\frac {2 a c \,\textit {\_a}^{1+n} m -\textit {\_a}^{m +1} b n +2 a c \,\textit {\_a}^{1+n}-\textit {\_a}^{m +1} b}{\left (1+n \right ) \left (m +1\right )}}}{\left (c +y \left (x \right )\right ) \textit {\_a}}d \textit {\_a} +\frac {{\mathrm e}^{-\frac {2 a c \,x^{1+n} m -x^{m +1} b n +2 a c \,x^{1+n}-x^{m +1} b}{\left (1+n \right ) \left (m +1\right )}}}{c +y \left (x \right )} = 0 \]
✓ Solution by Mathematica
Time used: 2.15 (sec). Leaf size: 286
DSolve[y'[x]==a*x^n*y[x]^2+b*x^m*y[x]+b*c*x^m-a*c^2*x^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {e^{\frac {b x^{m+1}}{m+1}-\frac {2 a c x^{n+1}}{n+1}}}{a b (m-n) (c+K[2])^2}-\int _1^x\left (-\frac {\exp \left (\frac {b K[1]^{m+1}}{m+1}-\frac {2 a c K[1]^{n+1}}{n+1}\right ) K[1]^n}{b (m-n) (c+K[2])}-\frac {\exp \left (\frac {b K[1]^{m+1}}{m+1}-\frac {2 a c K[1]^{n+1}}{n+1}\right ) \left (-b K[1]^m+a c K[1]^n-a K[2] K[1]^n\right )}{a b (m-n) (c+K[2])^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {\exp \left (\frac {b K[1]^{m+1}}{m+1}-\frac {2 a c K[1]^{n+1}}{n+1}\right ) \left (-b K[1]^m+a c K[1]^n-a y(x) K[1]^n\right )}{a b (m-n) (c+y(x))}dK[1]=c_1,y(x)\right ] \]