77.1.135 problem 162 (page 236)
Internal
problem
ID
[17954]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
162
(page
236)
Date
solved
:
Monday, March 31, 2025 at 04:52:20 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+9 y&=\ln \left (2 \sin \left (\frac {x}{2}\right )\right ) \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 125
ode:=diff(diff(y(x),x),x)+9*y(x) = ln(2*sin(1/2*x));
dsolve(ode,y(x), singsol=all);
\[
y = \sin \left (3 x \right ) c_2 +\cos \left (3 x \right ) c_1 -\frac {1}{54}-\frac {i \left (\operatorname {csgn}\left (-2 \sin \left (\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\frac {i x}{2}}\right )-1\right ) \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{i x}-1\right )\right )}{18}+\frac {i \operatorname {csgn}\left (i {\mathrm e}^{-\frac {i x}{2}}\right ) \pi }{18}-\frac {i \left (\operatorname {csgn}\left (-2 \sin \left (\frac {x}{2}\right )\right )+1\right ) \pi \,\operatorname {csgn}\left (i \sin \left (\frac {x}{2}\right )\right )}{18}+\frac {\left (1-\cos \left (3 x \right )\right ) \ln \left ({\mathrm e}^{i x}-1\right )}{9}+\frac {i x \,{\mathrm e}^{3 i x}}{18}-\frac {\ln \left ({\mathrm e}^{\frac {i x}{2}}\right )}{9}-\frac {\cos \left (2 x \right )}{9}-\frac {{\mathrm e}^{-i x}}{36}-\frac {{\mathrm e}^{i x}}{36}
\]
✓ Mathematica. Time used: 0.107 (sec). Leaf size: 76
ode=D[y[x],{x,2}]+9*y[x]==Log[2*Sin[x/2]];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {1}{54} \left (-3 x \sin (3 x)-3 \cos (x)-6 \cos (2 x)+10 \cos (3 x)+6 \log \left (2 \sin \left (\frac {x}{2}\right )\right )+54 c_1 \cos (3 x)+54 c_2 \sin (3 x)-6 \cos (3 x) \log \left (2 \sin \left (\frac {x}{2}\right )\right )-1\right )
\]
✓ Sympy. Time used: 107.777 (sec). Leaf size: 53
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(9*y(x) - log(2*sin(x/2)) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{1} - \frac {\int \left (\log {\left (\sin {\left (\frac {x}{2} \right )} \right )} + \log {\left (2 \right )}\right ) \sin {\left (3 x \right )}\, dx}{3}\right ) \cos {\left (3 x \right )} + \left (C_{2} + \frac {\int \left (\log {\left (\sin {\left (\frac {x}{2} \right )} \right )} + \log {\left (2 \right )}\right ) \cos {\left (3 x \right )}\, dx}{3}\right ) \sin {\left (3 x \right )}
\]