53.1.15 problem 15
Internal
problem
ID
[8449]
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
:
15
Date
solved
:
Monday, January 27, 2025 at 04:01:30 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (x^{2}+y^{2}\right )^{2} {y^{\prime }}^{2}&=4 x^{2} y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 251
dsolve((x^2+y(x)^2)^2*diff(y(x),x)^2=4*x^2*y(x)^2,y(x), singsol=all)
\begin{align*}
y &= \frac {1-\sqrt {4 c_{1}^{2} x^{2}+1}}{2 c_{1}} \\
y &= \frac {1+\sqrt {4 c_{1}^{2} x^{2}+1}}{2 c_{1}} \\
y &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{{1}/{3}}}{4 \sqrt {c_{1}}}-\frac {\sqrt {c_{1}}\, x^{2} \left (i \sqrt {3}-1\right )}{\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{{1}/{3}}} \\
y &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{{2}/{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{{2}/{3}}}{4 \left (4+4 \sqrt {4 c_{1}^{3} x^{6}+1}\right )^{{1}/{3}} \sqrt {c_{1}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 16.301 (sec). Leaf size: 345
DSolve[(x^2+y[x]^2)^2*(D[y[x],x])^2==4*x^2*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{2} \left (-\sqrt {4 x^2+e^{2 c_1}}-e^{c_1}\right ) \\
y(x)\to \frac {1}{2} \left (\sqrt {4 x^2+e^{2 c_1}}-e^{c_1}\right ) \\
y(x)\to \frac {\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \left (\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}+\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2}{4 \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 x^6+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\
y(x)\to 0 \\
\end{align*}