23.2.56 problem 58

Internal problem ID [5411]
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 : 58
Date solved : Tuesday, September 30, 2025 at 12:39:59 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}-2 a \,x^{3} y^{\prime }+4 a \,x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.363 (sec). Leaf size: 27

ode:=diff(y(x),x)^2-2*a*x^3*diff(y(x),x)+4*a*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.731 (sec). Leaf size: 175

ode=(D[y[x],x])^2-2*a*x^3*D[y[x],x]+4*a*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 {a} x \sqrt {a x^4-4 y(x)} \text {arctanh}\left (\frac {\sqrt {a x^4-4 y(x)}}{\sqrt {a} x^2}\right )}{2 \sqrt {a x^2 \left (a x^4-4 y(x)\right )}}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {\sqrt {a} x \sqrt {a x^4-4 y(x)} \text {arctanh}\left (\frac {\sqrt {a x^4-4 y(x)}}{\sqrt {a} x^2}\right )}{2 \sqrt {a x^2 \left (a x^4-4 y(x)\right )}}+\frac {1}{4} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to \frac {a x^4}{4} \end{align*}

✓ Sympy. Time used: 1.454 (sec). Leaf size: 12

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

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