Internal problem ID [4085]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 846.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {x {y^{\prime }}^{2}-2 y=-x} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 73
dsolve(x*diff(y(x),x)^2+x-2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \left (\frac {\left (\operatorname {LambertW}\left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )+1\right )^{2}}{2 \operatorname {LambertW}\left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}}+\frac {1}{2}\right ) x y \left (x \right ) = \left (\frac {\left (\operatorname {LambertW}\left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )+1\right )^{2}}{2 \operatorname {LambertW}\left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}}+\frac {1}{2}\right ) x \end{align*}
✓ Solution by Mathematica
Time used: 0.642 (sec). Leaf size: 97
DSolve[x (y'[x])^2+x-2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {2}{\sqrt {\frac {2 y(x)}{x}-1}-1}-2 \log \left (\sqrt {\frac {2 y(x)}{x}-1}-1\right )=\log (x)+c_1,y(x)\right ] \text {Solve}\left [\frac {2}{\sqrt {\frac {2 y(x)}{x}-1}+1}+2 \log \left (\sqrt {\frac {2 y(x)}{x}-1}+1\right )=-\log (x)+c_1,y(x)\right ] \end{align*}