23.2.249 problem 256

Internal problem ID [5604]
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 : 256
Date solved : Tuesday, September 30, 2025 at 01:11:39 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.141 (sec). Leaf size: 159

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

\begin{align*} y &= 2^{{3}/{4}} \left (-a x \right )^{{1}/{4}} \\ y &= -2^{{3}/{4}} \left (-a x \right )^{{1}/{4}} \\ y &= -i 2^{{3}/{4}} \left (-a x \right )^{{1}/{4}} \\ y &= i 2^{{3}/{4}} \left (-a x \right )^{{1}/{4}} \\ y &= \frac {2^{{1}/{4}} \left (a \left (c_1 -x \right )^{2} c_1^{3}\right )^{{1}/{4}}}{c_1} \\ y &= -\frac {2^{{1}/{4}} \left (a \left (c_1 -x \right )^{2} c_1^{3}\right )^{{1}/{4}}}{c_1} \\ y &= -\frac {i 2^{{1}/{4}} \left (a \left (c_1 -x \right )^{2} c_1^{3}\right )^{{1}/{4}}}{c_1} \\ y &= \frac {i 2^{{1}/{4}} \left (a \left (c_1 -x \right )^{2} c_1^{3}\right )^{{1}/{4}}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.951 (sec). Leaf size: 151

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

\begin{align*} y(x)&\to -\frac {e^{-\frac {c_1}{4}} \sqrt {-8 a x+e^{c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {e^{-\frac {c_1}{4}} \sqrt {-8 a x+e^{c_1}}}{\sqrt {2}}\\ y(x)&\to -(-2)^{3/4} \sqrt [4]{a} \sqrt [4]{x}\\ y(x)&\to (-2)^{3/4} \sqrt [4]{a} \sqrt [4]{x}\\ y(x)&\to (-1-i) \sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}\\ y(x)&\to (1+i) \sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x} \end{align*}

✗ Sympy

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

Timed Out