29.29.9 problem 831

Internal problem ID [5414]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 831
Date solved : Monday, January 27, 2025 at 11:21:32 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 31

dsolve(2*diff(y(x),x)^2+x*diff(y(x),x)-2*y(x) = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {x^{2} \left (1+2 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{4}}}{4}\right )\right )}{16 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{4}}}{4}\right )^{2}} \]

✓ Solution by Mathematica

Time used: 1.274 (sec). Leaf size: 126

DSolve[2 (D[y[x],x])^2+x D[y[x],x]-2 y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\frac {\frac {1}{2} x \sqrt {x^2+16 y(x)}+8 y(x) \log \left (\sqrt {x^2+16 y(x)}+x\right )-\frac {x^2}{2}}{8 y(x)}&=c_1,y(x)\right ] \\ \text {Solve}\left [\log (y(x))-\frac {\frac {1}{2} x \sqrt {x^2+16 y(x)}+8 y(x) \log \left (\sqrt {x^2+16 y(x)}+x\right )+\frac {x^2}{2}}{8 y(x)}&=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}