Internal problem ID [9619]
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]
Solve \begin {gather*} \boxed {y^{\prime }+a n \,x^{n -1} y^{2}-c \,x^{m} \left (x^{n} a +b \right ) y+c \,x^{m}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 397
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)
\[ y \relax (x ) = \frac {x^{n} \left (\int -\frac {{\mathrm e}^{\frac {c \,x^{m} x \left (x^{n} a m +a \,x^{n}+b m +b n +b \right )}{\left (m +1\right ) \left (m +n +1\right )}} a n \,x^{n}}{\left (a \,x^{n}+b \right )^{2} x}d x \right ) a +c_{1} a \,x^{n}+\left (\int -\frac {{\mathrm e}^{\frac {c \,x^{m} x \left (x^{n} a m +a \,x^{n}+b m +b n +b \right )}{\left (m +1\right ) \left (m +n +1\right )}} a n \,x^{n}}{\left (a \,x^{n}+b \right )^{2} x}d x \right ) b +c_{1} b -{\mathrm e}^{\frac {c \,x^{m} x \left (x^{n} a m +a \,x^{n}+b m +b n +b \right )}{\left (m +1\right ) \left (m +n +1\right )}}}{\left (\int -\frac {{\mathrm e}^{\frac {c \,x^{m} x \left (x^{n} a m +a \,x^{n}+b m +b n +b \right )}{\left (m +1\right ) \left (m +n +1\right )}} a n \,x^{n}}{\left (a \,x^{n}+b \right )^{2} x}d x \right ) x^{2 n} a^{2}+x^{2 n} c_{1} a^{2}+2 \left (\int -\frac {{\mathrm e}^{\frac {c \,x^{m} x \left (x^{n} a m +a \,x^{n}+b m +b n +b \right )}{\left (m +1\right ) \left (m +n +1\right )}} a n \,x^{n}}{\left (a \,x^{n}+b \right )^{2} x}d x \right ) x^{n} a b +2 x^{n} c_{1} a b +\left (\int -\frac {{\mathrm e}^{\frac {c \,x^{m} x \left (x^{n} a m +a \,x^{n}+b m +b n +b \right )}{\left (m +1\right ) \left (m +n +1\right )}} a n \,x^{n}}{\left (a \,x^{n}+b \right )^{2} x}d x \right ) b^{2}+c_{1} b^{2}} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
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]
Not solved