82.8.3 problem Ex. 3
Internal
problem
ID
[18680]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
II.
Equations
of
the
first
order
and
of
the
first
degree.
Exercises
at
page
25
Problem
number
:
Ex.
3
Date
solved
:
Monday, March 31, 2025 at 05:56:23 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 3 x^{2} y^{4}+2 x y+\left (2 x^{3} y^{3}-x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 271
ode:=3*x^2*y(x)^4+2*x*y(x)+(2*x^3*y(x)^3-x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12^{{1}/{3}} \left (c_1 12^{{1}/{3}} x +\left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_1^{3}}{x}}\right )^{{2}/{3}}\right )}{6 x^{2} \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_1^{3}}{x}}\right )^{{1}/{3}}} \\
y &= \frac {\left (\left (-i \sqrt {3}-1\right ) \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_1^{3}}{x}}\right )^{{2}/{3}}+\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x c_1 2^{{2}/{3}}\right ) 2^{{2}/{3}} 3^{{1}/{3}}}{12 \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_1^{3}}{x}}\right )^{{1}/{3}} x^{2}} \\
y &= -\frac {\left (\left (1-i \sqrt {3}\right ) \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_1^{3}}{x}}\right )^{{2}/{3}}+x \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_1 2^{{2}/{3}}\right ) 2^{{2}/{3}} 3^{{1}/{3}}}{12 \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_1^{3}}{x}}\right )^{{1}/{3}} x^{2}} \\
\end{align*}
✓ Mathematica. Time used: 60.262 (sec). Leaf size: 349
ode=(3*x^2*y[x]^4+2*x*y[x])+(2*x^3*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 {2 \sqrt [3]{3} e^{\frac {7 c_1}{3}} x^3+\sqrt [3]{2} \left (-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}\right ){}^{2/3}}{6^{2/3} x^3 \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}}{2 \sqrt [3]{2} 3^{2/3} x^3}-\frac {\left (\sqrt {3}+3 i\right ) e^{\frac {7 c_1}{3}}}{2^{2/3} 3^{5/6} \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}} \\
y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}}{2 \sqrt [3]{2} 3^{2/3} x^3}-\frac {\left (\sqrt {3}-3 i\right ) e^{\frac {7 c_1}{3}}}{2^{2/3} 3^{5/6} \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2*y(x)**4 + 2*x*y(x) + (2*x**3*y(x)**3 - x**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out