60.2.332 problem 909
Internal
problem
ID
[10919]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
909
Date
solved
:
Tuesday, January 28, 2025 at 05:31:02 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {x^{3}+y^{4} x^{3}+2 x^{2} y^{2}+x +x^{3} y^{6}+3 x^{2} y^{4}+3 x y^{2}+1}{x^{5} y} \end{align*}
✓ Solution by Maple
Time used: 2.306 (sec). Leaf size: 720
dsolve(diff(y(x),x) = (x^3+y(x)^4*x^3+2*x^2*y(x)^2+x+x^3*y(x)^6+3*x^2*y(x)^4+3*x*y(x)^2+1)/x^5/y(x),y(x), singsol=all)
\begin{align*}
y &= -\frac {2^{{1}/{3}} \sqrt {3}\, \sqrt {\left (\frac {2^{{2}/{3}} \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{2}/{3}}}{4}-\frac {2^{{1}/{3}} \left (x +3\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}}}{2}+x^{2}\right ) x \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}}}}{3 \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}} x} \\
y &= \frac {2^{{1}/{3}} \sqrt {3}\, \sqrt {\left (\frac {2^{{2}/{3}} \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{2}/{3}}}{4}-\frac {2^{{1}/{3}} \left (x +3\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}}}{2}+x^{2}\right ) x \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}}}}{3 \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}} x} \\
y &= -\frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {4}\, \sqrt {x \left (-\frac {2^{{2}/{3}} \left (1+i \sqrt {3}\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{{2}/{3}}}{4}-2^{{1}/{3}} \left (x +3\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{{1}/{3}}+x^{2} \left (i \sqrt {3}-1\right )\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{{1}/{3}}}}{12 \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{{1}/{3}} x} \\
y &= \frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {4}\, \sqrt {x \left (-\frac {2^{{2}/{3}} \left (1+i \sqrt {3}\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{{2}/{3}}}{4}-2^{{1}/{3}} \left (x +3\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{{1}/{3}}+x^{2} \left (i \sqrt {3}-1\right )\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{{1}/{3}}}}{12 \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{{1}/{3}} x} \\
y &= -\frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {-4 x \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}} \left (-\frac {2^{{2}/{3}} \left (i \sqrt {3}-1\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{2}/{3}}}{4}+2^{{1}/{3}} \left (x +3\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}}+x^{2} \left (1+i \sqrt {3}\right )\right )}}{12 \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}} x} \\
y &= \frac {2^{{5}/{6}} \sqrt {3}\, \sqrt {-4 x \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}} \left (-\frac {2^{{2}/{3}} \left (i \sqrt {3}-1\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{2}/{3}}}{4}+2^{{1}/{3}} \left (x +3\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}}+x^{2} \left (1+i \sqrt {3}\right )\right )}}{12 \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{{1}/{3}} x} \\
y &= \frac {\sqrt {x \left (\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+1}d \textit {\_a} \right ) x +c_{1} x +1\right ) x -1\right )}}{x} \\
y &= -\frac {\sqrt {x \left (\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+1}d \textit {\_a} \right ) x +c_{1} x +1\right ) x -1\right )}}{x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.125 (sec). Leaf size: 64
DSolve[D[y[x],x] == (1 + x + x^3 + 3*x*y[x]^2 + 2*x^2*y[x]^2 + 3*x^2*y[x]^4 + x^3*y[x]^4 + x^3*y[x]^6)/(x^5*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\frac {1}{2} \text {RootSum}\left [2 \text {$\#$1}^3+2 \text {$\#$1}^2+1\&,\frac {\log \left (\frac {x y(x)^2+1}{x}-\text {$\#$1}\right )}{3 \text {$\#$1}^2+2 \text {$\#$1}}\&\right ]+\frac {1}{x}+c_1=0,y(x)\right ]
\]