Internal problem ID [10621]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing
arbitrary functions (but not containing their derivatives).
Problem number: 28.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+a \ln \left (x \right ) y^{2}-a f \left (x \right ) \left (\ln \left (x \right ) x -x \right ) y=-f \left (x \right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 227
dsolve(diff(y(x),x)=-a*ln(x)*y(x)^2+a*f(x)*(x*ln(x)-x)*y(x)-f(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-x \left (\ln \left (x \right )-1\right ) {\mathrm e}^{\int \frac {f \left (x \right ) \ln \left (x \right )^{2} a \,x^{2}+\left (-2 x^{2} a f \left (x \right )-2\right ) \ln \left (x \right )+x^{2} a f \left (x \right )}{x \left (\ln \left (x \right )-1\right )}d x}+c_{1} a -\left (\int \ln \left (x \right ) {\mathrm e}^{a \left (\int \frac {x f \left (x \right ) \ln \left (x \right )^{2}}{\ln \left (x \right )-1}d x \right )-2 a \left (\int \frac {x f \left (x \right ) \ln \left (x \right )}{\ln \left (x \right )-1}d x \right )+a \left (\int \frac {x f \left (x \right )}{\ln \left (x \right )-1}d x \right )-2 \left (\int \frac {\ln \left (x \right )}{x \left (\ln \left (x \right )-1\right )}d x \right )}d x \right )}{a x \left (\ln \left (x \right )-1\right ) \left (c_{1} a -\left (\int \ln \left (x \right ) {\mathrm e}^{a \left (\int \frac {x f \left (x \right ) \ln \left (x \right )^{2}}{\ln \left (x \right )-1}d x \right )-2 a \left (\int \frac {x f \left (x \right ) \ln \left (x \right )}{\ln \left (x \right )-1}d x \right )+a \left (\int \frac {x f \left (x \right )}{\ln \left (x \right )-1}d x \right )-2 \left (\int \frac {\ln \left (x \right )}{x \left (\ln \left (x \right )-1\right )}d x \right )}d x \right )\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==-a*Log[x]*y[x]^2+a*f[x]*(x*Log[x]-x)*y[x]-f[x],y[x],x,IncludeSingularSolutions -> True]
Not solved