29.24.8 problem 670
Internal
problem
ID
[5261]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
670
Date
solved
:
Monday, January 27, 2025 at 10:50:35 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (1-x^{3}+6 x^{2} y^{2}\right ) y^{\prime }&=\left (6+3 y x -4 y^{3}\right ) x \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 484
dsolve((1-x^3+6*x^2*y(x)^2)*diff(y(x),x) = (6+3*x*y(x)-4*y(x)^3)*x,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {6 x^{3}+\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{{2}/{3}}-6}{6 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {6 i \sqrt {3}\, x^{3}-i \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{{2}/{3}} \sqrt {3}-6 x^{3}-6 i \sqrt {3}-\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{{2}/{3}}+6}{12 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {6 i \sqrt {3}\, x^{3}-i \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{{2}/{3}} \sqrt {3}+6 x^{3}-6 i \sqrt {3}+\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{{2}/{3}}-6}{12 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.117 (sec). Leaf size: 424
DSolve[(1-x^3+6 x^2 y[x]^2)D[y[x],x]==(6+3 x y[x]-4 y[x]^3)x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{2} \left (x^3-1\right )}{\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}-\frac {\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{6 \sqrt [3]{2} x^2} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2} \\
\end{align*}