28.1.118 problem 141
Internal
problem
ID
[4424]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
2.
First-Order
and
Simple
Higher-Order
Differential
Equations.
Page
78
Problem
number
:
141
Date
solved
:
Monday, January 27, 2025 at 09:16:49 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y^{4}+y x +\left (x y^{3}-x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.082 (sec). Leaf size: 338
dsolve((y(x)^4+x*y(x))+(x*y(x)^3-x^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{1}/{3}}+\frac {c_{1}^{2}}{\left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{1}/{3}}}+c_{1}}{6 x} \\
y \left (x \right ) &= \frac {i \sqrt {3}\, c_{1}^{2}-i \sqrt {3}\, \left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{2}/{3}}-c_{1}^{2}+2 c_{1} \left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{1}/{3}}-\left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{2}/{3}}}{12 x \left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )-\frac {\left (i c_{1} \sqrt {3}+c_{1} -2 \left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{1}/{3}}\right ) c_{1}}{\left (-54 x^{4}+6 \sqrt {81 x^{4}-3 c_{1}^{3}}\, x^{2}+c_{1}^{3}\right )^{{1}/{3}}}}{12 x} \\
\end{align*}
✓ Solution by Mathematica
Time used: 14.426 (sec). Leaf size: 355
DSolve[(y[x]^4+x*y[x])+(x*y[x]^3-x^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\frac {2\ 2^{2/3} c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+24 c_1{}^3 x^4}+4 c_1{}^3}}+\sqrt [3]{54 x^4+6 \sqrt {81 x^8+24 c_1{}^3 x^4}+8 c_1{}^3}+2 c_1}{6 x} \\
y(x)\to \frac {\frac {2\ 2^{2/3} \left (1+i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+24 c_1{}^3 x^4}+4 c_1{}^3}}+\left (1-i \sqrt {3}\right ) \sqrt [3]{54 x^4+6 \sqrt {81 x^8+24 c_1{}^3 x^4}+8 c_1{}^3}-4 c_1}{12 x} \\
y(x)\to \frac {\frac {2\ 2^{2/3} \left (1-i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+24 c_1{}^3 x^4}+4 c_1{}^3}}+\left (1+i \sqrt {3}\right ) \sqrt [3]{54 x^4+6 \sqrt {81 x^8+24 c_1{}^3 x^4}+8 c_1{}^3}-4 c_1}{12 x} \\
y(x)\to 0 \\
\end{align*}