47.3.1 problem 3

Internal problem ID [9749]
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 : 3
Date solved : Tuesday, September 30, 2025 at 06:32:47 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}+x^{3} y^{\prime }-2 x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.222 (sec). Leaf size: 21

ode:=diff(y(x),x)^2+x^3*diff(y(x),x)-2*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {x^{4}}{8} \\ y &= c_{1} \left (x^{2}+2 c_{1} \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.512 (sec). Leaf size: 142

ode=(D[y[x],x])^2+x^3*D[y[x],x]-2*x^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\frac {1}{4} \log (y(x))-\frac {\sqrt {x^6+8 x^2 y(x)} \text {arctanh}\left (\frac {\sqrt {x^4+8 y(x)}}{x^2}\right )}{2 x \sqrt {x^4+8 y(x)}}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {\sqrt {x^6+8 x^2 y(x)} \text {arctanh}\left (\frac {\sqrt {x^4+8 y(x)}}{x^2}\right )}{2 x \sqrt {x^4+8 y(x)}}+\frac {1}{4} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to -\frac {x^4}{8} \end{align*}

✓ Sympy. Time used: 1.159 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), x) - 2*x**2*y(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (C_{1} + x^{2}\right )}{2} \]