23.1.617 problem 611

Internal problem ID [5224]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 611
Date solved : Tuesday, September 30, 2025 at 11:56:25 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} \left (x^{4}+y^{2}\right ) y^{\prime }&=4 x^{3} y \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 41

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

\begin{align*} y &= -\frac {\sqrt {4 x^{4}+c_1^{2}}}{2}+\frac {c_1}{2} \\ y &= \frac {\sqrt {4 x^{4}+c_1^{2}}}{2}+\frac {c_1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 58

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

\begin{align*} y(x)&\to \frac {1}{2} \left (c_1-\sqrt {4 x^4+c_1{}^2}\right )\\ y(x)&\to \frac {1}{2} \left (\sqrt {4 x^4+c_1{}^2}+c_1\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.482 (sec). Leaf size: 34

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

\[ \left [ y{\left (x \right )} = 2 C_{1} - \sqrt {4 C_{1}^{2} + x^{4}}, \ y{\left (x \right )} = 2 C_{1} + \sqrt {4 C_{1}^{2} + x^{4}}\right ] \]