61.19.28 problem 28

Internal problem ID [12297]
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
Date solved : Tuesday, January 28, 2025 at 02:20:16 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-a \ln \left (x \right ) y^{2}+a f \left (x \right ) \left (x \ln \left (x \right )-x \right ) y-f \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.023 (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)
 

\[ y = \frac {-x \left (-1+\ln \left (x \right )\right ) {\mathrm e}^{\int \frac {\ln \left (x \right )^{2} f a \,x^{2}+\left (-2 f a \,x^{2}-2\right ) \ln \left (x \right )+f a \,x^{2}}{x \left (-1+\ln \left (x \right )\right )}d x}+c_{1} a -\int \ln \left (x \right ) {\mathrm e}^{a \left (\int \frac {x \ln \left (x \right )^{2} f}{-1+\ln \left (x \right )}d x \right )-2 a \left (\int \frac {x \ln \left (x \right ) f}{-1+\ln \left (x \right )}d x \right )+a \left (\int \frac {x f}{-1+\ln \left (x \right )}d x \right )-2 \left (\int \frac {\ln \left (x \right )}{x \left (-1+\ln \left (x \right )\right )}d x \right )}d x}{a x \left (-1+\ln \left (x \right )\right ) \left (c_{1} a -\int \ln \left (x \right ) {\mathrm e}^{a \left (\int \frac {x \ln \left (x \right )^{2} f}{-1+\ln \left (x \right )}d x \right )-2 a \left (\int \frac {x \ln \left (x \right ) f}{-1+\ln \left (x \right )}d x \right )+a \left (\int \frac {x f}{-1+\ln \left (x \right )}d x \right )-2 \left (\int \frac {\ln \left (x \right )}{x \left (-1+\ln \left (x \right )\right )}d x \right )}d x \right )} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[D[y[x],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