28.1.12 problem 12

Internal problem ID [4318]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 12
Date solved : Sunday, March 30, 2025 at 02:59:23 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.251 (sec). Leaf size: 115

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

\begin{align*} y &= \frac {\sqrt {-c_1 \,x^{2}-\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= \frac {\sqrt {-c_1 \,x^{2}+\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= -\frac {\sqrt {-c_1 \,x^{2}-\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= -\frac {\sqrt {-c_1 \,x^{2}+\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ \end{align*}

✓ Mathematica. Time used: 8.613 (sec). Leaf size: 218

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

\begin{align*} y(x)\to -\sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to 0 \\ y(x)\to -\sqrt {-\sqrt {x^4}-x^2} \\ y(x)\to \sqrt {-\sqrt {x^4}-x^2} \\ y(x)\to -\sqrt {\sqrt {x^4}-x^2} \\ y(x)\to \sqrt {\sqrt {x^4}-x^2} \\ \end{align*}

✓ Sympy. Time used: 3.603 (sec). Leaf size: 75

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

\[ \left [ y{\left (x \right )} = - \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = - \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}\right ] \]