15.12.21 problem 21
Internal
problem
ID
[3165]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
20,
page
90
Problem
number
:
21
Date
solved
:
Sunday, March 30, 2025 at 01:20:08 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+9 y&=\csc \left (2 x \right ) \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 77
ode:=diff(diff(y(x),x),x)+9*y(x) = csc(2*x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\sin \left (x \right ) \left (4 \cos \left (x \right )^{2}-1\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{6}+\frac {\left (4 \cos \left (x \right )^{3}-3 \cos \left (x \right )\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{6}+4 c_1 \cos \left (x \right )^{3}+4 \sin \left (x \right ) \cos \left (x \right )^{2} c_2 +\frac {\left (-9 c_1 +4 \sin \left (x \right )\right ) \cos \left (x \right )}{3}-\sin \left (x \right ) c_2
\]
✓ Mathematica. Time used: 0.038 (sec). Leaf size: 65
ode=D[y[x],{x,2}]+9*y[x]==Csc[2*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {1}{6} \left (\cos (3 x) \text {arctanh}(\sin (x))+4 \sin (2 x)+\sin (3 x) \log \left (\sin \left (\frac {x}{2}\right )\right )+6 c_1 \cos (3 x)+6 c_2 \sin (3 x)-\sin (3 x) \log \left (\cos \left (\frac {x}{2}\right )\right )\right )
\]
✓ Sympy. Time used: 1.151 (sec). Leaf size: 63
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(9*y(x) + Derivative(y(x), (x, 2)) - 1/sin(2*x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{1} - \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{12} + \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{12} - \frac {2 \sin {\left (x \right )}}{3}\right ) \cos {\left (3 x \right )} + \left (C_{2} + \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{12} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{12} + \frac {2 \cos {\left (x \right )}}{3}\right ) \sin {\left (3 x \right )}
\]