70.17.9 problem 18
Internal
problem
ID
[18981]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
4.
Second
order
linear
equations.
Section
4.7
(Variation
of
parameters).
Problems
at
page
280
Problem
number
:
18
Date
solved
:
Thursday, October 02, 2025 at 03:36:37 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} 4 y^{\prime \prime }+y&=2 \sec \left (2 t \right ) \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 169
ode:=4*diff(diff(y(t),t),t)+y(t) = 2*sec(2*t);
dsolve(ode,y(t), singsol=all);
\[
y = \frac {\sqrt {2-\sqrt {2}}\, \sin \left (\frac {t}{2}\right ) \left (1+\sqrt {2}\right ) \operatorname {arctanh}\left (\sin \left (\frac {t}{2}\right ) \sqrt {2-\sqrt {2}}\, \left (2+\sqrt {2}\right )\right )}{2}+\frac {\sqrt {2+\sqrt {2}}\, \cos \left (\frac {t}{2}\right ) \left (\sqrt {2}-1\right ) \operatorname {arctanh}\left (\cos \left (\frac {t}{2}\right ) \sqrt {2+\sqrt {2}}\, \left (\sqrt {2}-2\right )\right )}{2}+\frac {\sqrt {2-\sqrt {2}}\, \cos \left (\frac {t}{2}\right ) \left (1+\sqrt {2}\right ) \operatorname {arctanh}\left (\cos \left (\frac {t}{2}\right ) \sqrt {2-\sqrt {2}}\, \left (2+\sqrt {2}\right )\right )}{2}+\frac {\sqrt {2+\sqrt {2}}\, \sin \left (\frac {t}{2}\right ) \left (\sqrt {2}-1\right ) \operatorname {arctanh}\left (\sin \left (\frac {t}{2}\right ) \sqrt {2+\sqrt {2}}\, \left (\sqrt {2}-2\right )\right )}{2}+\cos \left (\frac {t}{2}\right ) c_1 +\sin \left (\frac {t}{2}\right ) c_2
\]
✓ Mathematica. Time used: 0.193 (sec). Leaf size: 136
ode=4*D[y[t],{t,2}]+y[t]==2*Sec[2*t];
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \cos \left (\frac {t}{2}\right ) \int _1^t-\sec (2 K[1]) \sin \left (\frac {K[1]}{2}\right )dK[1]+\frac {1}{2} \sqrt {2+\sqrt {2}} \sin \left (\frac {t}{2}\right ) \text {arctanh}\left (\frac {2 \sin \left (\frac {t}{2}\right )}{\sqrt {2-\sqrt {2}}}\right )-\frac {\sin \left (\frac {t}{2}\right ) \text {arctanh}\left (\frac {2 \sin \left (\frac {t}{2}\right )}{\sqrt {2+\sqrt {2}}}\right )}{\sqrt {2 \left (2+\sqrt {2}\right )}}+c_1 \cos \left (\frac {t}{2}\right )+c_2 \sin \left (\frac {t}{2}\right ) \end{align*}
✓ Sympy. Time used: 1.043 (sec). Leaf size: 39
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(y(t) + 4*Derivative(y(t), (t, 2)) - 2/cos(2*t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \left (C_{1} - \int \frac {\sin {\left (\frac {t}{2} \right )}}{\cos {\left (2 t \right )}}\, dt\right ) \cos {\left (\frac {t}{2} \right )} + \left (C_{2} + \int \frac {\cos {\left (\frac {t}{2} \right )}}{\cos {\left (2 t \right )}}\, dt\right ) \sin {\left (\frac {t}{2} \right )}
\]