35.2.6 problem 6

Internal problem ID [6098]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 2. Separable equations. page 398
Problem number : 6
Date solved : Wednesday, March 05, 2025 at 12:12:28 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (\sqrt {2}\right )&=0 \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 31

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

\begin{align*} y &= -\frac {\sqrt {2 x^{2}-4}\, x}{2} \\ y &= \frac {\sqrt {2 x^{2}-4}\, x}{2} \\ \end{align*}

✓ Mathematica. Time used: 3.914 (sec). Leaf size: 48

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

\begin{align*} y(x)\to -\frac {\sqrt {x^2 \left (x^2-2\right )}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {x^2 \left (x^2-2\right )}}{\sqrt {2}} \\ \end{align*}

✓ Sympy. Time used: 0.892 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {x^{4} - 2 x^{2}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {x^{4} - 2 x^{2}}}{2}\right ] \]