29.25.15 problem 712
Internal
problem
ID
[5302]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
25
Problem
number
:
712
Date
solved
:
Monday, January 27, 2025 at 11:06:10 AM
CAS
classification
:
[_rational]
\begin{align*} \left (x +2 y+2 x^{2} y^{3}+x y^{4}\right ) y^{\prime }+\left (1+y^{4}\right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 629
dsolve((x+2*y(x)+2*x^2*y(x)^3+x*y(x)^4)*diff(y(x),x)+(1+y(x)^4)*y(x) = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {-1+\frac {\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+4 c_{1} x^{4}+18 c_{1}^{2} x^{2}-x^{2}-4 c_{1}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{1}/{3}}}{2}-\frac {2 \left (3 c_{1} x^{2}-1\right )}{\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+4 c_{1} x^{4}+18 c_{1}^{2} x^{2}-x^{2}-4 c_{1}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{1}/{3}}}}{3 c_{1} x} \\
y \left (x \right ) &= \frac {i \left (4-12 c_{1} x^{2}-\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{2}/{3}}\right ) \sqrt {3}+12 c_{1} x^{2}-{\left (\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{1}/{3}}+2\right )}^{2}}{12 \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{1}/{3}} x c_{1}} \\
y \left (x \right ) &= \frac {12 i \sqrt {3}\, c_{1} x^{2}+i \sqrt {3}\, \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{2}/{3}}+12 c_{1} x^{2}-4 i \sqrt {3}-\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{2}/{3}}-4 \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{1}/{3}}-4}{12 c_{1} x \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 8.854 (sec). Leaf size: 675
DSolve[(x+2 y[x]+2 x^2 y[x]^3+x y[x]^4)D[y[x],x]+(1+y[x]^4)y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\frac {2 c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+2^{2/3} \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+2 c_1}{6 x} \\
y(x)\to \frac {-\frac {2 i \left (\sqrt {3}-i\right ) c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x} \\
y(x)\to \frac {\frac {2 i \left (\sqrt {3}+i\right ) c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x} \\
y(x)\to 0 \\
y(x)\to -\sqrt [4]{-1} \\
y(x)\to \sqrt [4]{-1} \\
y(x)\to -(-1)^{3/4} \\
y(x)\to (-1)^{3/4} \\
y(x)\to \frac {1}{2} x \left (-1+\frac {i x^2}{\sqrt {-x^4}}\right ) \\
y(x)\to -\frac {x}{2}+\frac {i \sqrt {-x^4}}{2 x} \\
\end{align*}