23.2.173 problem 177

Internal problem ID [5528]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 177
Date solved : Tuesday, September 30, 2025 at 12:50:30 PM
CAS classification : [_linear]

\begin{align*} 4 x^{2} {y^{\prime }}^{2}-4 x y y^{\prime }&=8 x^{3}-y^{2} \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 30

ode:=4*x^2*diff(y(x),x)^2-4*x*y(x)*diff(y(x),x) = 8*x^3-y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left (-\sqrt {2}\, x +c_1 \right ) \sqrt {x} \\ y &= \left (\sqrt {2}\, x +c_1 \right ) \sqrt {x} \\ \end{align*}

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 42

ode=4 x^2 (D[y[x],x])^2-4 x y[x] D[y[x],x]==8 x^3 -y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt {x} \left (-\sqrt {2} x+c_1\right )\\ y(x)&\to \sqrt {x} \left (\sqrt {2} x+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.317 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**3 + 4*x**2*Derivative(y(x), x)**2 - 4*x*y(x)*Derivative(y(x), x) + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \sqrt {x} \left (C_{1} - \sqrt {2} x\right ), \ y{\left (x \right )} = \sqrt {x} \left (C_{1} + \sqrt {2} x\right )\right ] \]