Internal problem ID [9878]
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]
Solve \begin {gather*} \boxed {y^{\prime }+a \ln \relax (x ) y^{2}-a f \relax (x ) \left (x \ln \relax (x )-x \right ) y+f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 348
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)
\[ y \relax (x ) = \frac {x \,{\mathrm e}^{\int \frac {\ln \relax (x )^{2} f \relax (x ) a \,x^{2}-2 \ln \relax (x ) f \relax (x ) a \,x^{2}+f \relax (x ) a \,x^{2}-2 \ln \relax (x )}{x \left (\ln \relax (x )-1\right )}d x} \ln \relax (x )-x \,{\mathrm e}^{\int \frac {\ln \relax (x )^{2} f \relax (x ) a \,x^{2}-2 \ln \relax (x ) f \relax (x ) a \,x^{2}+f \relax (x ) a \,x^{2}-2 \ln \relax (x )}{x \left (\ln \relax (x )-1\right )}d x}-c_{1} a +\int \ln \relax (x ) {\mathrm e}^{a \left (\int \frac {x \ln \relax (x )^{2} f \relax (x )}{\ln \relax (x )-1}d x \right )-2 a \left (\int \frac {x \ln \relax (x ) f \relax (x )}{\ln \relax (x )-1}d x \right )+a \left (\int \frac {x f \relax (x )}{\ln \relax (x )-1}d x \right )-2 \left (\int \frac {\ln \relax (x )}{x \left (\ln \relax (x )-1\right )}d x \right )}d x}{a x \left (-\ln \relax (x ) c_{1} a +\left (\int \ln \relax (x ) {\mathrm e}^{a \left (\int \frac {x \ln \relax (x )^{2} f \relax (x )}{\ln \relax (x )-1}d x \right )-2 a \left (\int \frac {x \ln \relax (x ) f \relax (x )}{\ln \relax (x )-1}d x \right )+a \left (\int \frac {x f \relax (x )}{\ln \relax (x )-1}d x \right )-2 \left (\int \frac {\ln \relax (x )}{x \left (\ln \relax (x )-1\right )}d x \right )}d x \right ) \ln \relax (x )+c_{1} a -\left (\int \ln \relax (x ) {\mathrm e}^{a \left (\int \frac {x \ln \relax (x )^{2} f \relax (x )}{\ln \relax (x )-1}d x \right )-2 a \left (\int \frac {x \ln \relax (x ) f \relax (x )}{\ln \relax (x )-1}d x \right )+a \left (\int \frac {x f \relax (x )}{\ln \relax (x )-1}d x \right )-2 \left (\int \frac {\ln \relax (x )}{x \left (\ln \relax (x )-1\right )}d x \right )}d x \right )\right )} \]
✗ 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