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 : Sunday, March 30, 2025 at 01:11:38 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 31

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

\[ y = \frac {x^{2} \left (1+2 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_1}{4}}}{4}\right )\right )}{16 \operatorname {LambertW}\left (\frac {x \,{\mathrm e}^{\frac {c_1}{4}}}{4}\right )^{2}} \]

✓ Mathematica. Time used: 1.261 (sec). Leaf size: 126

ode=2*(D[y[x],x])^2+x*D[y[x],x]-2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) - 2*y(x) + 2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x/4 - sqrt(x**2 + 16*y(x))/4 + Derivative(y(x), x) cannot be solved by the factorable group method