32.10.24 problem Exercise 35.23(c), page 504

Internal problem ID [6018]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 8. Special second order equations. Lesson 35. Independent variable x absent
Problem number : Exercise 35.23(c), page 504
Date solved : Tuesday, January 28, 2025 at 03:09:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 22

dsolve(x*y(x)*diff(y(x),x$2)-2*x*(diff(y(x),x))^2+(1+y(x))*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= c_1 \tanh \left (\frac {\ln \left (x \right )-c_2}{2 c_1}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 22.975 (sec). Leaf size: 52

DSolve[x*y[x]*D[y[x],{x,2}]-2*x*(D[y[x],x])^2+(1+y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\tan \left (\frac {\sqrt {c_1} (\log (x)-c_2)}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {c_1}} \\ y(x)\to \frac {1}{2} (\log (x)-c_2) \\ \end{align*}