Internal problem ID [2324]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 37, page 171
Problem number: 11.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {y-y^{\prime } x \left (1+y^{\prime }\right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 69
dsolve(y(x)=diff(y(x),x)*x*(1+diff(y(x),x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {x \left (\frac {1}{2 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )}+1\right )}{2 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )} y \left (x \right ) = \frac {x \left (\frac {1}{2 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )}+1\right )}{2 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )} \end{align*}
✓ Solution by Mathematica
Time used: 0.566 (sec). Leaf size: 102
DSolve[y[x]==y'[x]*x*(1+y'[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {1}{\sqrt {\frac {4 y(x)}{x}+1}-1}-\log \left (\sqrt {\frac {4 y(x)}{x}+1}-1\right )=\frac {\log (x)}{2}+c_1,y(x)\right ] \text {Solve}\left [\frac {1}{\sqrt {\frac {4 y(x)}{x}+1}+1}+\log \left (\sqrt {\frac {4 y(x)}{x}+1}+1\right )=-\frac {\log (x)}{2}+c_1,y(x)\right ] y(x)\to 0 \end{align*}