29.25.11 problem 708
Internal
problem
ID
[5298]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
25
Problem
number
:
708
Date
solved
:
Monday, January 27, 2025 at 11:04:58 AM
CAS
classification
:
[_rational]
\begin{align*} \left (a^{2} x^{2}+\left (x^{2}+y^{2}\right )^{2}\right ) y^{\prime }&=a^{2} x y \end{align*}
✓ Solution by Maple
Time used: 0.034 (sec). Leaf size: 197
dsolve((a^2*x^2+(x^2+y(x)^2)^2)*diff(y(x),x) = a^2*x*y(x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {\sqrt {-2 a^{2}-2 x^{2}-2 \sqrt {x^{4}+\left (2 a^{2}-2 c_{1} \right ) x^{2}+\left (a^{2}+c_{1} \right )^{2}}-2 c_{1}}}{2} \\
y \left (x \right ) &= \frac {\sqrt {-2 a^{2}-2 x^{2}-2 \sqrt {x^{4}+\left (2 a^{2}-2 c_{1} \right ) x^{2}+\left (a^{2}+c_{1} \right )^{2}}-2 c_{1}}}{2} \\
y \left (x \right ) &= -\frac {\sqrt {-2 a^{2}-2 x^{2}+2 \sqrt {x^{4}+\left (2 a^{2}-2 c_{1} \right ) x^{2}+\left (a^{2}+c_{1} \right )^{2}}-2 c_{1}}}{2} \\
y \left (x \right ) &= \frac {\sqrt {-2 a^{2}-2 x^{2}+2 \sqrt {x^{4}+\left (2 a^{2}-2 c_{1} \right ) x^{2}+\left (a^{2}+c_{1} \right )^{2}}-2 c_{1}}}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 6.521 (sec). Leaf size: 272
DSolve[(a^2 x^2+(x^2+y[x]^2)^2)D[y[x],x]==a^2 x y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {-\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {-\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\
y(x)\to -\frac {\sqrt {\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\
y(x)\to \frac {\sqrt {\sqrt {\left (a^2+x^2-c_1{}^2\right ){}^2+4 c_1{}^2 x^2}-a^2-x^2+c_1{}^2}}{\sqrt {2}} \\
y(x)\to 0 \\
y(x)\to -\sqrt {-x^2} \\
y(x)\to \sqrt {-x^2} \\
\end{align*}