12.1.4 problem Problem 14.3 (a)

Internal problem ID [3460]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 14, First order ordinary differential equations. 14.4 Exercises, page 490
Problem number : Problem 14.3 (a)
Date solved : Tuesday, September 30, 2025 at 06:39:02 AM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y \left (2 x^{2} y^{2}+1\right ) y^{\prime }+x \left (y^{4}+1\right )&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 119

ode:=y(x)*(2*x^2*y(x)^2+1)*diff(y(x),x)+x*(y(x)^4+1) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {-2-2 \sqrt {-4 x^{4}-8 c_1 \,x^{2}+1}}}{2 x} \\ y &= \frac {\sqrt {-2-2 \sqrt {-4 x^{4}-8 c_1 \,x^{2}+1}}}{2 x} \\ y &= -\frac {\sqrt {2}\, \sqrt {-1+\sqrt {-4 x^{4}-8 c_1 \,x^{2}+1}}}{2 x} \\ y &= \frac {\sqrt {2}\, \sqrt {-1+\sqrt {-4 x^{4}-8 c_1 \,x^{2}+1}}}{2 x} \\ \end{align*}

✓ Mathematica. Time used: 11.906 (sec). Leaf size: 197

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

\begin{align*} y(x)&\to -\frac {\sqrt {-\frac {1+\sqrt {-4 x^4+8 c_1 x^2+1}}{x^2}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-\frac {1+\sqrt {-4 x^4+8 c_1 x^2+1}}{x^2}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\frac {-1+\sqrt {-4 x^4+8 c_1 x^2+1}}{x^2}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\frac {-1+\sqrt {-4 x^4+8 c_1 x^2+1}}{x^2}}}{\sqrt {2}}\\ y(x)&\to -\sqrt [4]{-1}\\ y(x)&\to \sqrt [4]{-1}\\ y(x)&\to -(-1)^{3/4}\\ y(x)&\to (-1)^{3/4} \end{align*}

✓ Sympy. Time used: 4.266 (sec). Leaf size: 136

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

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