18.1.4 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, March 04, 2025 at 04:40:04 PM
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: 0.017 (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 \left (x \right ) &= -\frac {\sqrt {-2-2 \sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\
y \left (x \right ) &= \frac {\sqrt {-2-2 \sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\
y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-1+\sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\
y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-1+\sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\
\end{align*}
✓ Mathematica. Time used: 13.911 (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: 5.757 (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)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {\frac {- \sqrt {C_{1} x^{2} - 4 x^{4} + 1} - 1}{x^{2}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {\frac {- \sqrt {C_{1} x^{2} - 4 x^{4} + 1} - 1}{x^{2}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {\frac {\sqrt {C_{1} x^{2} - 4 x^{4} + 1} - 1}{x^{2}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {\frac {\sqrt {C_{1} x^{2} - 4 x^{4} + 1} - 1}{x^{2}}}}{2}\right ]
\]