61.7.9 problem 9

Internal problem ID [12160]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-1. Equations Containing Logarithmic Functions
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 12:45:52 AM
CAS classification : [_Riccati]

\begin{align*} x^{2} \ln \left (a x \right ) \left (y^{\prime }-y^{2}\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 45

dsolve(x^2*ln(a*x)*(diff(y(x),x)-y(x)^2)=1,y(x), singsol=all)
 

\[ y = \frac {-c_{1} \operatorname {Ei}_{1}\left (-\ln \left (a x \right )\right )+1}{x \left (\left (c_{1} \operatorname {Ei}_{1}\left (-\ln \left (a x \right )\right )-1\right ) \ln \left (a x \right )+c_{1} a x \right )} \]

✓ Solution by Mathematica

Time used: 0.754 (sec). Leaf size: 102

DSolve[x^2*Log[a*x]*(D[y[x],x]-y[x]^2)==1,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {c_1 \log (a x) \int _1^x\frac {1}{\log ^2(a K[1])}dK[1]+\log (a x)+c_1 x}{\log ^2(a x) \left (x+c_1 x \int _1^x\frac {1}{\log ^2(a K[1])}dK[1]\right )} \\ y(x)\to -\frac {\frac {1}{\int _1^x\frac {1}{\log ^2(a K[1])}dK[1]}+\frac {\log (a x)}{x}}{\log ^2(a x)} \\ \end{align*}