53.3.18 problem 21

Internal problem ID [8480]
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 equation. EXERCISES Page 320
Problem number : 21
Date solved : Monday, January 27, 2025 at 04:07:07 PM
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: 0.026 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 1.261 (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*}