Internal problem ID [10362]
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: 32.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+a n \,x^{n -1} y^{2}-c \,x^{m} \left (x^{n} a +b \right ) y=-c \,x^{m}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 199
dsolve(diff(y(x),x)=-a*n*x^(n-1)*y(x)^2+c*x^m*(a*x^n+b)*y(x)-c*x^m,y(x), singsol=all)
\[ y \left (x \right ) = \frac {a n \left (a \,x^{n}+b \right ) \left (\int \frac {x^{n -1} {\mathrm e}^{\frac {c \left (a \left (1+m \right ) x^{1+m +n}+b \,x^{1+m} \left (1+m +n \right )\right )}{\left (1+m \right ) \left (1+m +n \right )}}}{\left (a \,x^{n}+b \right )^{2}}d x \right )-x^{n} c_{1} a -c_{1} b +{\mathrm e}^{\frac {c \left (a \left (1+m \right ) x^{1+m +n}+b \,x^{1+m} \left (1+m +n \right )\right )}{\left (1+m \right ) \left (1+m +n \right )}}}{\left (a \left (\int \frac {x^{n -1} {\mathrm e}^{\frac {x \left (a \left (1+m \right ) x^{n}+b \left (1+m +n \right )\right ) c \,x^{m}}{\left (1+m \right ) \left (1+m +n \right )}}}{\left (a \,x^{n}+b \right )^{2}}d x \right ) n -c_{1} \right ) \left (a^{2} x^{2 n}+2 x^{n} a b +b^{2}\right )} \]
✓ Solution by Mathematica
Time used: 8.659 (sec). Leaf size: 304
DSolve[y'[x]==-a*n*x^(n-1)*y[x]^2+c*x^m*(a*x^n+b)*y[x]-c*x^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a c_1 n \left (a x^n+b\right ) \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]+a^2 n x^n+c_1 e^{c x^{m+1} \left (\frac {a x^n}{m+n+1}+\frac {b}{m+1}\right )}+a b n}{a n \left (a x^n+b\right )^2 \left (1+c_1 \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]\right )} \\ y(x)\to \frac {\frac {e^{c x^{m+1} \left (\frac {a x^n}{m+n+1}+\frac {b}{m+1}\right )}}{a n \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]}+a x^n+b}{\left (a x^n+b\right )^2} \\ \end{align*}