Internal problem ID [10361]
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: 31.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}+a \,x^{n} \left (b \,x^{m}+c \right ) y=b m \,x^{m -1}} \]
✗ Solution by Maple
dsolve(diff(y(x),x)=a*x^n*y(x)^2-a*x^n*(b*x^m+c)*y(x)+b*m*x^(m-1),y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 59.342 (sec). Leaf size: 353
DSolve[y'[x]==a*x^n*y[x]^2-a*x^n*(b*x^m+c)*y[x]+b*m*x^(m-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b m \left (b x^m+c\right )^2 \left (1+c_1 \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]\right )}{b c_1 m \left (b x^m+c\right ) \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]+c_1 e^{a x^{n+1} \left (\frac {b x^m}{m+n+1}+\frac {c}{n+1}\right )}+b^2 m x^m+b c m} \\ y(x)\to \frac {b m \left (b x^m+c\right )^2 \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]}{b m \left (b x^m+c\right ) \int _1^x\frac {\exp \left (a K[1]^{n+1} \left (\frac {b K[1]^m}{m+n+1}+\frac {c}{n+1}\right )\right ) K[1]^{m-1}}{\left (b K[1]^m+c\right )^2}dK[1]+e^{a x^{n+1} \left (\frac {b x^m}{m+n+1}+\frac {c}{n+1}\right )}} \\ \end{align*}