29.21.29 problem 605
Internal
problem
ID
[5199]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
21
Problem
number
:
605
Date
solved
:
Sunday, March 30, 2025 at 06:51:45 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+b^{2}+x^{2}+2 x y&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 565
ode:=(a^2+x^2+y(x)^2)*diff(y(x),x)+b^2+x^2+2*x*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {2 \left (a^{2}+x^{2}-\frac {\left (-12 b^{2} x -4 x^{3}-12 c_1 +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 c_1 \,b^{2} x +6 c_1 \,x^{3}+9 c_1^{2}}\right )^{{2}/{3}}}{4}\right )}{\left (-12 b^{2} x -4 x^{3}-12 c_1 +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 c_1 \,b^{2} x +6 c_1 \,x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-12 b^{2} x -4 x^{3}-12 c_1 +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 c_1 \,b^{2} x +6 c_1 \,x^{3}+9 c_1^{2}}\right )^{{2}/{3}}+\left (a^{2}+x^{2}\right ) \left (i \sqrt {3}-1\right )}{\left (-12 b^{2} x -4 x^{3}-12 c_1 +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 c_1 \,b^{2} x +6 c_1 \,x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-12 b^{2} x -4 x^{3}-12 c_1 +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 c_1 \,b^{2} x +6 c_1 \,x^{3}+9 c_1^{2}}\right )^{{2}/{3}}}{4}+\left (a^{2}+x^{2}\right ) \left (1+i \sqrt {3}\right )}{\left (-12 b^{2} x -4 x^{3}-12 c_1 +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 c_1 \,b^{2} x +6 c_1 \,x^{3}+9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 7.771 (sec). Leaf size: 438
ode=(a^2+x^2+y[x]^2)D[y[x],x]+b^2+x^2+2 x y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} \left (\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1\right ){}^{2/3}-2 a^2-2 x^2}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (a^2+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}{2 \sqrt [3]{2}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (a^2+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}{2 \sqrt [3]{2}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(b**2 + x**2 + 2*x*y(x) + (a**2 + x**2 + y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out