80.1.8 problem 8(b)
Internal
problem
ID
[21126]
Book
:
A
Textbook
on
Ordinary
Differential
Equations
by
Shair
Ahmad
and
Antonio
Ambrosetti.
Second
edition.
ISBN
978-3-319-16407-6.
Springer
2015
Section
:
Chapter
1.
First
order
linear
differential
equations.
Excercise
1.5
at
page
13
Problem
number
:
8(b)
Date
solved
:
Thursday, October 02, 2025 at 07:09:27 PM
CAS
classification
:
[_linear]
\begin{align*} x^{\prime }+\sec \left (t \right ) x&=\frac {1}{t -1} \end{align*}
With initial conditions
\begin{align*}
x \left (\frac {\pi }{4}\right )&=1 \\
\end{align*}
✓ Maple. Time used: 0.119 (sec). Leaf size: 36
ode:=diff(x(t),t)+sec(t)*x(t) = 1/(t-1);
ic:=[x(1/4*Pi) = 1];
dsolve([ode,op(ic)],x(t), singsol=all);
\[
x = \frac {\int _{\frac {\pi }{4}}^{t}\frac {\sec \left (\textit {\_z1} \right )+\tan \left (\textit {\_z1} \right )}{\textit {\_z1} -1}d \textit {\_z1} +1+\sqrt {2}}{\sec \left (t \right )+\tan \left (t \right )}
\]
✓ Mathematica. Time used: 27.85 (sec). Leaf size: 46
ode=D[x[t],t]+Sec[t]*x[t]==1/(t-1);
ic={x[Pi/4]==1};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to e^{-\coth ^{-1}(\sin (t))} \left (\int _{\frac {\pi }{4}}^t\frac {e^{\coth ^{-1}(\sin (K[1]))}}{K[1]-1}dK[1]+e^{\coth ^{-1}\left (\frac {1}{\sqrt {2}}\right )}\right ) \end{align*}
✓ Sympy. Time used: 77.974 (sec). Leaf size: 241
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(x(t)*sec(t) + Derivative(x(t), t) - 1/(t - 1),0)
ics = {x(pi/4): 1}
dsolve(ode,func=x(t),ics=ics)
\[
x{\left (t \right )} = \frac {\sqrt {\sin {\left (t \right )} - 1} \left (\int \limits ^{\frac {\pi }{4}} \frac {\sqrt {\sin {\left (t \right )} + 1}}{t \sqrt {\sin {\left (t \right )} - 1} - \sqrt {\sin {\left (t \right )} - 1}}\, dt + \int \limits ^{\frac {\pi }{4}} \frac {\sqrt {\sin {\left (t \right )} + 1} \sec {\left (t \right )}}{\sqrt {\sin {\left (t \right )} - 1}}\, dt + \int \frac {\sqrt {\sin {\left (t \right )} + 1} \left (t x{\left (t \right )} \sec {\left (t \right )} - x{\left (t \right )} \sec {\left (t \right )} - 1\right )}{\left (t - 1\right ) \sqrt {\sin {\left (t \right )} - 1}}\, dt + \int \limits ^{\frac {\pi }{4}} \frac {\sqrt {\sin {\left (t \right )} + 1} x{\left (t \right )} \sec {\left (t \right )}}{t \sqrt {\sin {\left (t \right )} - 1} - \sqrt {\sin {\left (t \right )} - 1}}\, dt - \int \limits ^{\frac {\pi }{4}} \frac {t \sqrt {\sin {\left (t \right )} + 1} x{\left (t \right )} \sec {\left (t \right )}}{t \sqrt {\sin {\left (t \right )} - 1} - \sqrt {\sin {\left (t \right )} - 1}}\, dt + \frac {i \sqrt {\sqrt {2} + 2}}{\sqrt {2 - \sqrt {2}}}\right )}{\sqrt {\sin {\left (t \right )} - 1} \int \frac {\sqrt {\sin {\left (t \right )} + 1} \sec {\left (t \right )}}{\sqrt {\sin {\left (t \right )} - 1}}\, dt - \sqrt {\sin {\left (t \right )} + 1}}
\]