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]
\[ \boxed {{y^{\prime }}^{2} x -2 y=-x} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 96
dsolve(x*diff(y(x),x)^2+x-2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (2 \operatorname {LambertW}\left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}+2 \operatorname {LambertW}\left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )+1\right ) x}{2 \operatorname {LambertW}\left (\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}} \\ y \left (x \right ) &= \frac {\left (2 \operatorname {LambertW}\left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )+1\right ) x}{2 \operatorname {LambertW}\left (-\frac {\sqrt {c_{1} x}}{c_{1}}\right )^{2}} \\ \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*}