12.5.27 problem 24

Internal problem ID [1651]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 24
Date solved : Tuesday, March 04, 2025 at 01:05:06 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

With initial conditions

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 18

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

\[ y = \frac {\sqrt {-2 x^{4}+18}}{2 x} \]

✓ Mathematica. Time used: 0.192 (sec). Leaf size: 25

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

\[ y(x)\to \frac {\sqrt {9-x^4}}{\sqrt {2} x} \]

✓ Sympy. Time used: 0.433 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \frac {\sqrt {18 - 2 x^{4}}}{2 x} \]