61.8.8 problem 17

Internal problem ID [12168]
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 : 17
Date solved : Tuesday, January 28, 2025 at 12:46:21 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Riccati]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 32

dsolve(x*diff(y(x),x)=(a*y(x)+b*ln(x))^2,y(x), singsol=all)
 

\[ y = \frac {-\ln \left (x \right ) a b +\tan \left (\left (\ln \left (x \right )+c_{1} \right ) \sqrt {a b}\right ) \sqrt {a b}}{a^{2}} \]

✓ Solution by Mathematica

Time used: 4.308 (sec). Leaf size: 43

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

\[ y(x)\to -\frac {b \log (x)}{a}+\sqrt {\frac {b}{a^3}} \tan \left (a^2 \sqrt {\frac {b}{a^3}} \log (x)+c_1\right ) \]