82.12.7 problem Ex. 7
Internal
problem
ID
[18693]
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.
Examples
on
chapter
II
at
page
29
Problem
number
:
Ex.
7
Date
solved
:
Thursday, March 13, 2025 at 12:38:50 PM
CAS
classification
:
[_rational, _Bernoulli]
\begin{align*} 3 y^{\prime }+\frac {2 y}{x +1}&=\frac {x^{3}}{y^{2}} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 135
ode:=3*diff(y(x),x)+2/(1+x)*y(x) = x^3/y(x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {450^{{1}/{3}} {\left (\left (10 x^{6}+24 x^{5}+15 x^{4}+60 c_{1} \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}}}{30 \left (x +1\right )^{2}} \\
y \left (x \right ) &= -\frac {450^{{1}/{3}} {\left (\left (10 x^{6}+24 x^{5}+15 x^{4}+60 c_{1} \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{60 \left (x +1\right )^{2}} \\
y \left (x \right ) &= \frac {450^{{1}/{3}} {\left (\left (10 x^{6}+24 x^{5}+15 x^{4}+60 c_{1} \right ) \left (x +1\right )^{4}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{60 \left (x +1\right )^{2}} \\
\end{align*}
✓ Mathematica. Time used: 4.164 (sec). Leaf size: 144
ode=3*D[y[x],x]+2/(x+1)*y[x]==x^3/y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{\frac {10 x^6+24 x^5+15 x^4+60 c_1}{(x+1)^2}}}{2^{2/3} \sqrt [3]{15}} \\
y(x)\to -\frac {\sqrt [3]{-\frac {1}{15}} \sqrt [3]{\frac {10 x^6+24 x^5+15 x^4+60 c_1}{(x+1)^2}}}{2^{2/3}} \\
y(x)\to \frac {(-1)^{2/3} \sqrt [3]{\frac {10 x^6+24 x^5+15 x^4+60 c_1}{(x+1)^2}}}{2^{2/3} \sqrt [3]{15}} \\
\end{align*}
✓ Sympy. Time used: 3.118 (sec). Leaf size: 138
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**3/y(x)**2 + 3*Derivative(y(x), x) + 2*y(x)/(x + 1),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt [3]{\frac {C_{1} + 30 x^{6} + 72 x^{5} + 45 x^{4}}{x^{2} + 2 x + 1}} \left (- \sqrt [3]{150} - 3^{\frac {5}{6}} \sqrt [3]{50} i\right )}{60}, \ y{\left (x \right )} = \frac {\sqrt [3]{\frac {C_{1} + 30 x^{6} + 72 x^{5} + 45 x^{4}}{x^{2} + 2 x + 1}} \left (- \sqrt [3]{150} + 3^{\frac {5}{6}} \sqrt [3]{50} i\right )}{60}, \ y{\left (x \right )} = \frac {\sqrt [3]{150} \sqrt [3]{\frac {C_{1} + 30 x^{6} + 72 x^{5} + 45 x^{4}}{x^{2} + 2 x + 1}}}{30}\right ]
\]