38.2.39 problem 39

Internal problem ID [6468]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 24. First order differential equations. Further problems 24. page 1068
Problem number : 39
Date solved : Sunday, March 30, 2025 at 11:04:35 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

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

\begin{align*} y &= -\frac {\sqrt {3}\, \sqrt {x \left (x^{3}+3 c_1 \right )}}{3 x} \\ y &= \frac {\sqrt {3}\, \sqrt {x \left (x^{3}+3 c_1 \right )}}{3 x} \\ \end{align*}

✓ Mathematica. Time used: 0.21 (sec). Leaf size: 56

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

\begin{align*} y(x)\to -\frac {\sqrt {x^3+3 c_1}}{\sqrt {3} \sqrt {x}} \\ y(x)\to \frac {\sqrt {x^3+3 c_1}}{\sqrt {3} \sqrt {x}} \\ \end{align*}

✓ Sympy. Time used: 0.450 (sec). Leaf size: 39

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {3} \sqrt {\frac {C_{1}}{x} + x^{2}}}{3}, \ y{\left (x \right )} = \frac {\sqrt {3} \sqrt {\frac {C_{1}}{x} + x^{2}}}{3}\right ] \]