56.13.4 problem Ex 4

Internal problem ID [14160]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 24. Equations solvable for \(p\). Page 49
Problem number : Ex 4
Date solved : Thursday, October 02, 2025 at 09:17:52 AM
CAS classification : [_linear]

\begin{align*} \left (2 x y^{\prime }-y\right )^{2}&=8 x^{3} \end{align*}

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

ode:=(2*x*diff(y(x),x)-y(x))^2 = 8*x^3; 
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: 0.053 (sec). Leaf size: 42

ode=(2*x*D[y[x],x]-y[x])^2==8*x^3; 
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: 0.319 (sec). Leaf size: 31

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

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