Internal problem ID [9610]
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: 23.
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 ) \left (y^{\prime }-y^{2}\right )+a n \left (n -1\right ) x^{n -2}+b m \left (m -1\right ) x^{m -2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 173
dsolve((a*x^n+b*x^m+c)*(diff(y(x),x)-y(x)^2)+a*n*(n-1)*x^(n-2)+b*m*(m-1)*x^(m-2)=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (a^{2} x^{2 n} n +a b m \,x^{m +n}+a b n \,x^{m +n}+x^{2 m} b^{2} m +x^{n} a c n +x^{m} b c m \right ) \left (\int \frac {1}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2}}d x \right )+x^{2 n} c_{1} a^{2} n +x^{m +n} c_{1} a b m +x^{m +n} c_{1} a b n +x^{2 m} c_{1} b^{2} m +x^{n} c_{1} a c n +x^{m} c_{1} b c m +x}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2} x \left (c_{1}+\int \frac {1}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2}}d x \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(a*x^n+b*x^m+c)*(y'[x]-y[x]^2)+a*n*(n-1)*x^(n-2)+b*m*(m-1)*x^(m-2)==0,y[x],x,IncludeSingularSolutions -> True]
Not solved