Internal problem ID [2347]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 38, page 173
Problem number: 10.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {y-y^{\prime } x \left (y^{\prime }+1\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 65
dsolve(y(x)=diff(y(x),x)*x*(diff(y(x),x)+1),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {x \left (1+2 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right )}{4 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\ y \left (x \right ) &= \frac {x \left (1+2 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right )}{4 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.523 (sec). Leaf size: 102
DSolve[y[x]==y'[x]*x*(y'[x]+1),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*}