29.29.24 problem 846

Internal problem ID [5429]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 846
Date solved : Monday, January 27, 2025 at 11:23:21 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}+x -2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.049 (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: 0.583 (sec). Leaf size: 97

DSolve[x (D[y[x],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*}