29.25.6 problem 703
Internal
problem
ID
[5293]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
25
Problem
number
:
703
Date
solved
:
Monday, January 27, 2025 at 11:04:43 AM
CAS
classification
:
[_rational]
\begin{align*} x \left (a +b x y^{3}\right ) y^{\prime }+\left (a +c \,x^{3} y\right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 488
dsolve(x*(a+b*x*y(x)^3)*diff(y(x),x)+(a+c*x^3*y(x))*y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {3^{{1}/{3}} \left (-x^{2} b \left (c \,x^{2}-2 c_{1} \right ) 3^{{1}/{3}}+{\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{2}/{3}}\right )}{3 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{1}/{3}} b x} \\
y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{2}/{3}}+x^{2} b \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) \left (c \,x^{2}-2 c_{1} \right )\right ) 3^{{1}/{3}}}{6 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{1}/{3}} b x} \\
y \left (x \right ) &= \frac {3^{{1}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{2}/{3}}+x^{2} b \left (c \,x^{2}-2 c_{1} \right ) \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right )\right )}{6 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{{1}/{3}} b x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 53.988 (sec). Leaf size: 484
DSolve[x(a+b x y[x]^3)D[y[x],x]+(a+c x^3 y[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {x \left (-c x^2+2 c_1\right )}{\sqrt [3]{3} \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}+\frac {\sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}{3^{2/3} b x} \\
y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}+\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) b x^2 \left (c x^2-2 c_1\right )}{6 b x \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}} \\
y(x)\to \frac {\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) b x^2 \left (c x^2-2 c_1\right )-i \sqrt [3]{3} \left (\sqrt {3}-i\right ) \left (9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}}{6 b x \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}} \\
\end{align*}