15.8.15 problem 15
Internal
problem
ID
[3018]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
12,
page
46
Problem
number
:
15
Date
solved
:
Monday, January 27, 2025 at 07:08:49 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 2 y x +y^{4}+\left (x y^{3}-2 x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.080 (sec). Leaf size: 326
dsolve((2*x*y(x)+y(x)^4)+(x*y(x)^3-2*x^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\frac {\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}}}{2}+\frac {2 c_{1}^{2}}{\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}}}+c_{1}}{3 x} \\
y &= \frac {\left (-1-i \sqrt {3}\right ) \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{2}/{3}}+4 c_{1} \left (i c_{1} \sqrt {3}-c_{1} +\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}}\right )}{12 \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}} x} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{2}/{3}}+4 \left (-i c_{1} \sqrt {3}-c_{1} +\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}}\right ) c_{1}}{12 \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{{1}/{3}} x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 17.179 (sec). Leaf size: 371
DSolve[(2*x*y[x]+y[x]^4)+(x*y[x]^3-2*x^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\frac {2 \sqrt [3]{2} c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}+2 c_1}{6 x} \\
y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}-4 c_1}{12 x} \\
y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}-4 c_1}{12 x} \\
y(x)\to 0 \\
\end{align*}