53.1.17 problem 17
Internal
problem
ID
[8451]
Book
:
Elementary
differential
equations.
By
Earl
D.
Rainville,
Phillip
E.
Bedient.
Macmilliam
Publishing
Co.
NY.
6th
edition.
1981.
Section
:
CHAPTER
16.
Nonlinear
equations.
Section
94.
Factoring
the
left
member.
EXERCISES
Page
309
Problem
number
:
17
Date
solved
:
Sunday, March 30, 2025 at 01:05:20 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _Bernoulli]
\begin{align*} x y \left (x^{2}+y^{2}\right ) \left ({y^{\prime }}^{2}-1\right )&=y^{\prime } \left (x^{4}+x^{2} y^{2}+y^{4}\right ) \end{align*}
✓ Maple. Time used: 0.275 (sec). Leaf size: 142
ode:=x*y(x)*(x^2+y(x)^2)*(diff(y(x),x)^2-1) = diff(y(x),x)*(x^4+x^2*y(x)^2+y(x)^4);
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}} \\
y &= \sqrt {2 \ln \left (x \right )+c_1}\, x \\
y &= -\sqrt {2 \ln \left (x \right )+c_1}\, x \\
\end{align*}
✓ Mathematica. Time used: 9.43 (sec). Leaf size: 248
ode=x*y[x]*(x^2+y[x]^2)*((D[y[x],x])^2-1)==D[y[x],x]*(x^4+x^2*y[x]^2+y[x]^4);
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 -x \sqrt {2 \log (x)+c_1} \\
y(x)\to x \sqrt {2 \log (x)+c_1} \\
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: 8.153 (sec). Leaf size: 104
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x**2 + y(x)**2)*(Derivative(y(x), x)**2 - 1)*y(x) - (x**4 + x**2*y(x)**2 + y(x)**4)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - x \sqrt {C_{1} + 2 \log {\left (x \right )}}, \ y{\left (x \right )} = x \sqrt {C_{1} + 2 \log {\left (x \right )}}, \ y{\left (x \right )} = - \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = - \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}\right ]
\]