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
:
Sunday, March 30, 2025 at 03:21:28 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y^{4}+x y+\left (x y^{3}-x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.030 (sec). Leaf size: 338
ode:=y(x)^4+x*y(x)+(x*y(x)^3-x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{1}/{3}}+\frac {c_1^{2}}{\left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{1}/{3}}}+c_1}{6 x} \\
y &= \frac {i \sqrt {3}\, c_1^{2}-i \left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{2}/{3}} \sqrt {3}-c_1^{2}+2 c_1 \left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{1}/{3}}-\left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{2}/{3}}}{12 x \left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{1}/{3}}-\frac {c_1 \left (i c_1 \sqrt {3}+c_1 -2 \left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{1}/{3}}\right )}{\left (-54 x^{4}+c_1^{3}+6 \sqrt {81 x^{4}-3 c_1^{3}}\, x^{2}\right )^{{1}/{3}}}}{12 x} \\
\end{align*}
✓ Mathematica. Time used: 14.426 (sec). Leaf size: 355
ode=(y[x]^4+x*y[x])+(x*y[x]^3-x^2)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*y(x) + (-x**2 + x*y(x)**3)*Derivative(y(x), x) + y(x)**4,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out