Internal problem ID [3582]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 851.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+y^{\prime } x -y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 69
dsolve(x*diff(y(x),x)^2+x*diff(y(x),x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (\frac {1}{4 \LambertW \left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}}+\frac {1}{2 \LambertW \left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )}\right ) x \\ y \relax (x ) = \left (\frac {1}{4 \LambertW \left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}}+\frac {1}{2 \LambertW \left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )}\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.568 (sec). Leaf size: 102
DSolve[x (y'[x])^2+x y'[x]-y[x]==0,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*}