Internal problem ID [5332]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page
65
Problem number: 26.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {x {y^{\prime }}^{2}-y y^{\prime }-y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 32
dsolve(x*diff(y(x),x)^2-y(x)*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*} y = 0 y = \frac {\left (\operatorname {LambertW}\left (\frac {x \,{\mathrm e}}{c_{1}}\right )-1\right )^{2} x}{\operatorname {LambertW}\left (\frac {x \,{\mathrm e}}{c_{1}}\right )} \end{align*}
✓ Solution by Mathematica
Time used: 2.255 (sec). Leaf size: 158
DSolve[x*y'[x]^2-y[x]*y'[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {y(x)}{4 x}+\frac {1}{4} \sqrt {\frac {y(x)}{x}} \sqrt {\frac {y(x)}{x}+4}-\log \left (\sqrt {\frac {y(x)}{x}+4}-\sqrt {\frac {y(x)}{x}}\right )=-\frac {\log (x)}{2}+c_1,y(x)\right ] \text {Solve}\left [\frac {1}{4} \left (\frac {y(x)}{x}+\sqrt {\frac {y(x)}{x}} \sqrt {\frac {y(x)}{x}+4}-4 \log \left (\sqrt {\frac {y(x)}{x}+4}-\sqrt {\frac {y(x)}{x}}\right )\right )=\frac {\log (x)}{2}+c_1,y(x)\right ] y(x)\to 0 \end{align*}