Internal problem ID [6059]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 99. Clairaut’s equation. EXERCISES Page
320
Problem number: 21.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {2 \left (y^{\prime }\right )^{2}+x y^{\prime }-2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.166 (sec). Leaf size: 36
dsolve(2*diff(y(x),x)^2+x*diff(y(x),x)-2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{2 \LambertW \left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{4}}}{4}\right )-\frac {c_{1}}{2}}+\frac {{\mathrm e}^{\LambertW \left (\frac {x \,{\mathrm e}^{\frac {c_{1}}{4}}}{4}\right )-\frac {c_{1}}{4}} x}{2} \]
✓ Solution by Mathematica
Time used: 2.405 (sec). Leaf size: 138
DSolve[2*(y'[x])^2+x*y'[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) \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+16 y(x)}}\right )-\frac {x^2}{2}}{8 y(x)}+\frac {1}{2} \log (y(x))=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {1}{2} \log (y(x))-\frac {\frac {1}{2} x \sqrt {x^2+16 y(x)}+8 y(x) \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+16 y(x)}}\right )+\frac {x^2}{2}}{8 y(x)}=c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}