89.31.7 problem 7

Internal problem ID [24952]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Exercises at page 246
Problem number : 7
Date solved : Thursday, October 02, 2025 at 11:36:31 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 2 x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }-1&=0 \end{align*}

✓ Maple. Time used: 0.359 (sec). Leaf size: 151

ode:=2*x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)-1 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.756 (sec). Leaf size: 129

ode=2*x*y[x]^2*D[y[x],x]^2-y[x]^3*D[y[x],x]-1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*y(x)**2*Derivative(y(x), x)**2 - y(x)**3*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out