Internal problem ID [8754]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 418.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {x {y^{\prime }}^{2}-y y^{\prime }+a y=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 42
dsolve(x*diff(y(x),x)^2-y(x)*diff(y(x),x)+a*y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\left (\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}}{c_{1} a}\right )-1\right )^{2} a x}{\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}}{c_{1} a}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.88 (sec). Leaf size: 173
DSolve[a*y[x] - y[x]*y'[x] + x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {-\sqrt {\frac {y(x)}{x}} \sqrt {\frac {y(x)}{x}-4 a}-4 a \log \left (\sqrt {\frac {y(x)}{x}-4 a}-\sqrt {\frac {y(x)}{x}}\right )+\frac {y(x)}{x}}{4 a}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \text {Solve}\left [-\frac {\sqrt {\frac {y(x)}{x}} \sqrt {\frac {y(x)}{x}-4 a}+4 a \log \left (\sqrt {\frac {y(x)}{x}-4 a}-\sqrt {\frac {y(x)}{x}}\right )+\frac {y(x)}{x}}{4 a}&=\frac {\log (x)}{2}+c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}