74.4.3 problem 3
Internal
problem
ID
[19840]
Book
:
Elementary
Differential
Equations.
By
Thornton
C.
Fry.
D
Van
Nostrand.
NY.
First
Edition
(1929)
Section
:
Chapter
IV.
Methods
of
solution:
First
order
equations.
section
31.
Problems
at
page
85
Problem
number
:
3
Date
solved
:
Thursday, October 02, 2025 at 04:47:35 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }+\frac {y}{x}&=\frac {\sin \left (x \right )}{y^{3}} \end{align*}
✓ Maple. Time used: 0.019 (sec). Leaf size: 156
ode:=diff(y(x),x)+y(x)/x = sin(x)/y(x)^3;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {{\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_1 \right )}^{{1}/{4}}}{x} \\
y &= -\frac {{\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_1 \right )}^{{1}/{4}}}{x} \\
y &= -\frac {i {\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_1 \right )}^{{1}/{4}}}{x} \\
y &= \frac {i {\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_1 \right )}^{{1}/{4}}}{x} \\
\end{align*}
✓ Mathematica. Time used: 0.338 (sec). Leaf size: 114
ode=D[y[x],x]+y[x]/x==Sin[x]/y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{9 \left (x^2-2\right ) \sin (x)-3 x \left (x^2-6\right ) \cos (x)+c_1}}{x}\\ y(x)&\to -\frac {\sqrt [3]{-1} \sqrt [3]{9 \left (x^2-2\right ) \sin (x)-3 x \left (x^2-6\right ) \cos (x)+c_1}}{x}\\ y(x)&\to \frac {(-1)^{2/3} \sqrt [3]{9 \left (x^2-2\right ) \sin (x)-3 x \left (x^2-6\right ) \cos (x)+c_1}}{x} \end{align*}
✓ Sympy. Time used: 3.384 (sec). Leaf size: 197
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - sin(x)/y(x)**3 + y(x)/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - i \sqrt [4]{\frac {C_{1}}{x^{4}} - 4 \cos {\left (x \right )} + \frac {16 \sin {\left (x \right )}}{x} + \frac {48 \cos {\left (x \right )}}{x^{2}} - \frac {96 \sin {\left (x \right )}}{x^{3}} - \frac {96 \cos {\left (x \right )}}{x^{4}}}, \ y{\left (x \right )} = i \sqrt [4]{\frac {C_{1}}{x^{4}} - 4 \cos {\left (x \right )} + \frac {16 \sin {\left (x \right )}}{x} + \frac {48 \cos {\left (x \right )}}{x^{2}} - \frac {96 \sin {\left (x \right )}}{x^{3}} - \frac {96 \cos {\left (x \right )}}{x^{4}}}, \ y{\left (x \right )} = - \sqrt [4]{\frac {C_{1}}{x^{4}} - 4 \cos {\left (x \right )} + \frac {16 \sin {\left (x \right )}}{x} + \frac {48 \cos {\left (x \right )}}{x^{2}} - \frac {96 \sin {\left (x \right )}}{x^{3}} - \frac {96 \cos {\left (x \right )}}{x^{4}}}, \ y{\left (x \right )} = \sqrt [4]{\frac {C_{1}}{x^{4}} - 4 \cos {\left (x \right )} + \frac {16 \sin {\left (x \right )}}{x} + \frac {48 \cos {\left (x \right )}}{x^{2}} - \frac {96 \sin {\left (x \right )}}{x^{3}} - \frac {96 \cos {\left (x \right )}}{x^{4}}}\right ]
\]