29.25.7 problem 704
Internal
problem
ID
[5294]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
25
Problem
number
:
704
Date
solved
:
Monday, January 27, 2025 at 11:04:45 AM
CAS
classification
:
[_rational]
\begin{align*} x \left (1-2 x^{2} y^{3}\right ) y^{\prime }+\left (1-2 x^{3} y^{2}\right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 809
dsolve(x*(1-2*x^2*y(x)^3)*diff(y(x),x)+(1-2*x^3*y(x)^2)*y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {{\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}}}{6 x}+\frac {\left (c_{1} -2 x \right )^{2} x}{6 {\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}}}+\frac {c_{1}}{6}-\frac {x}{3} \\
y \left (x \right ) &= \frac {-2 \left (-c_{1} x +2 x^{2}\right ) {\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}}-i \left (-c_{1}^{2} x^{2}+4 c_{1} x^{3}-4 x^{4}+{\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{2}/{3}}\right ) \sqrt {3}-4 x^{4}+4 c_{1} x^{3}-c_{1}^{2} x^{2}-{\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{2}/{3}}}{12 {\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}} x} \\
y \left (x \right ) &= \frac {2 \left (c_{1} x -2 x^{2}\right ) {\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}}+i \left (-c_{1}^{2} x^{2}+4 c_{1} x^{3}-4 x^{4}+{\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{2}/{3}}\right ) \sqrt {3}-4 x^{4}+4 c_{1} x^{3}-c_{1}^{2} x^{2}-{\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{2}/{3}}}{12 {\left (\left (c_{1}^{3} x^{2}-6 x^{3} c_{1}^{2}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 x^{3} c_{1}^{2}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{{1}/{3}} x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.112 (sec). Leaf size: 672
DSolve[x(1-2 x^2 y[x]^3)D[y[x],x]+(1-2 x^3 y[x]^2)y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {-2 x^3+c_1 x^2+\frac {x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}+\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{6 x^2} \\
y(x)\to \frac {2 x^2 (-2 x+c_1)-\frac {i \left (\sqrt {3}-i\right ) x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{12 x^2} \\
y(x)\to \frac {2 x^2 (-2 x+c_1)+\frac {i \left (\sqrt {3}+i\right ) x^4 (-2 x+c_1){}^2}{\sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-8 x^9+12 c_1 x^8-6 c_1{}^2 x^7+c_1{}^3 x^6-27 x^4+3 \sqrt {3} \sqrt {x^8 \left (16 x^5-24 c_1 x^4+12 c_1{}^2 x^3-2 c_1{}^3 x^2+27\right )}}}{12 x^2} \\
\end{align*}