Internal problem ID [9741]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-2
Problem number: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (1+n \right ) x^{n} y^{2}-a \,x^{1+n} \ln \left (x \right )^{m} y+a \ln \left (x \right )^{m}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 201
dsolve(diff(y(x),x)=-(n+1)*x^n*y(x)^2+a*x^(n+1)*(ln(x))^m*y(x)-a*(ln(x))^m,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-{\mathrm e}^{\int \frac {\ln \left (x \right )^{m} x^{n} a \,x^{2}-2 n -2}{x}d x} x^{n} x +\int \left (-x^{n} n \,{\mathrm e}^{a \left (\int x^{1+n} \ln \left (x \right )^{m}d x \right )-2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}-x^{n} {\mathrm e}^{a \left (\int x^{1+n} \ln \left (x \right )^{m}d x \right )-2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1} \right ) x^{-n}}{x \left (\int \left (-x^{n} n \,{\mathrm e}^{a \left (\int x^{1+n} \ln \left (x \right )^{m}d x \right )-2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}-x^{n} {\mathrm e}^{a \left (\int x^{1+n} \ln \left (x \right )^{m}d x \right )-2 n \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1} \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==-(n+1)*x^n*y[x]^2+a*x^(n+1)*(Log[x])^m*y[x]-a*(Log[x])^m,y[x],x,IncludeSingularSolutions -> True]
Not solved