29.24.9 problem 671
Internal
problem
ID
[5262]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
24
Problem
number
:
671
Date
solved
:
Monday, January 27, 2025 at 10:50:37 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} x \left (3+5 x -12 x y^{2}+4 x^{2} y\right ) y^{\prime }+\left (3+10 x -8 x y^{2}+6 x^{2} y\right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 735
dsolve(x*(3+5*x-12*x*y(x)^2+4*x^2*y(x))*diff(y(x),x)+(3+10*x-8*x*y(x)^2+6*x^2*y(x))*y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {2^{{2}/{3}} {\left (\left (2 x^{5}+45 x^{3}+3 \sqrt {3}\, \sqrt {8 c_{1} x^{5}-25 x^{6}-30 x^{5}+180 c_{1} x^{3}-509 x^{4}+108 c_{1} x^{2}-900 x^{3}+108 c_{1}^{2}-540 x^{2}-108 x}+27 x^{2}+54 c_{1} \right ) x \right )}^{{1}/{3}}}{12 x}+\frac {\left (x^{3}+15 x +9\right ) 2^{{1}/{3}}}{6 {\left (\left (2 x^{5}+45 x^{3}+3 \sqrt {3}\, \sqrt {8 c_{1} x^{5}-25 x^{6}-30 x^{5}+180 c_{1} x^{3}-509 x^{4}+108 c_{1} x^{2}-900 x^{3}+108 c_{1}^{2}-540 x^{2}-108 x}+27 x^{2}+54 c_{1} \right ) x \right )}^{{1}/{3}}}+\frac {x}{6} \\
y \left (x \right ) &= -\frac {-4 x^{2} \left (2 x^{6}+45 x^{4}+27 x^{3}+3 \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}\, x +54 c_{1} x \right )^{{1}/{3}}-2 x \left (i \sqrt {3}-1\right ) \left (x^{3}+15 x +9\right ) 2^{{1}/{3}}+2^{{2}/{3}} \left (1+i \sqrt {3}\right ) \left (2 x^{6}+45 x^{4}+27 x^{3}+3 \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}\, x +54 c_{1} x \right )^{{2}/{3}}}{24 \left (2 x^{6}+45 x^{4}+27 x^{3}+3 \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}\, x +54 c_{1} x \right )^{{1}/{3}} x} \\
y \left (x \right ) &= \frac {4 x^{2} \left (2 x^{6}+45 x^{4}+27 x^{3}+3 \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}\, x +54 c_{1} x \right )^{{1}/{3}}-2 x \left (1+i \sqrt {3}\right ) \left (x^{3}+15 x +9\right ) 2^{{1}/{3}}+2^{{2}/{3}} \left (i \sqrt {3}-1\right ) \left (2 x^{6}+45 x^{4}+27 x^{3}+3 \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}\, x +54 c_{1} x \right )^{{2}/{3}}}{24 \left (2 x^{6}+45 x^{4}+27 x^{3}+3 \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}\, x +54 c_{1} x \right )^{{1}/{3}} x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.185 (sec). Leaf size: 660
DSolve[x(3+5 x-12 x y[x]^2+4 x^2 y[x])D[y[x],x]+(3+10 x-8 x y[x]^2+6 x^2 y[x])y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{-16 x^9-360 x^7-216 x^6+432 c_1 x^4+8 \sqrt {-4 x^9 \left (x^3+15 x+9\right )^3+\left (2 x^9+45 x^7+27 x^6-54 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2}-\frac {\left (x^3+15 x+9\right ) x}{3\ 2^{2/3} \sqrt [3]{-2 x^9-45 x^7-27 x^6+54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (25 x^6+(30+8 c_1) x^5+509 x^4+180 (5+c_1) x^3+108 (5+c_1) x^2+108 x-108 c_1{}^2\right )}}}+\frac {x}{6} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-16 x^9-360 x^7-216 x^6+432 c_1 x^4+8 \sqrt {-4 x^9 \left (x^3+15 x+9\right )^3+\left (2 x^9+45 x^7+27 x^6-54 c_1 x^4\right ){}^2}}}{24 \sqrt [3]{2} x^2}+\frac {\left (1+i \sqrt {3}\right ) \left (x^3+15 x+9\right ) x}{6\ 2^{2/3} \sqrt [3]{-2 x^9-45 x^7-27 x^6+54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (25 x^6+(30+8 c_1) x^5+509 x^4+180 (5+c_1) x^3+108 (5+c_1) x^2+108 x-108 c_1{}^2\right )}}}+\frac {x}{6} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-16 x^9-360 x^7-216 x^6+432 c_1 x^4+8 \sqrt {-4 x^9 \left (x^3+15 x+9\right )^3+\left (2 x^9+45 x^7+27 x^6-54 c_1 x^4\right ){}^2}}}{24 \sqrt [3]{2} x^2}+\frac {\left (1-i \sqrt {3}\right ) \left (x^3+15 x+9\right ) x}{6\ 2^{2/3} \sqrt [3]{-2 x^9-45 x^7-27 x^6+54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (25 x^6+(30+8 c_1) x^5+509 x^4+180 (5+c_1) x^3+108 (5+c_1) x^2+108 x-108 c_1{}^2\right )}}}+\frac {x}{6} \\
\end{align*}