89.10.30 problem 31

Internal problem ID [24523]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 77
Problem number : 31
Date solved : Thursday, October 02, 2025 at 10:45:25 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.158 (sec). Leaf size: 23

ode:=x^4-4*x^2*y(x)^2-y(x)^4+4*x^3*y(x)*diff(y(x),x) = 0; 
ic:=[y(1) = 2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {\sqrt {-9 x^{2}+25}\, x}{3 x -5} \]

✓ Mathematica. Time used: 3.871 (sec). Leaf size: 28

ode=( x^4-4*x^2*y[x]^2 -y[x]^4 )+( 4*x^3*y[x] )*D[y[x],x]==0; 
ic={y[1]==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt {x^2 (3 x+5)}}{\sqrt {5-3 x}} \end{align*}

✓ Sympy. Time used: 1.111 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4 + 4*x**3*y(x)*Derivative(y(x), x) - 4*x**2*y(x)**2 - y(x)**4,0) 
ics = {y(1): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \sqrt {\frac {- x - \frac {5}{3}}{x - \frac {5}{3}}} \]