68.3.23 problem 18

Internal problem ID [17170]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 18
Date solved : Thursday, October 02, 2025 at 01:49:04 PM
CAS classification : [_linear]

\begin{align*} y^{\prime }+y \sec \left (t \right )&=t \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 49

ode:=diff(y(t),t)+y(t)*sec(t) = t; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {\left (i \pi ^{2}+12 i t^{2}+48 i \operatorname {polylog}\left (2, -i {\mathrm e}^{i t}\right )-48 t \ln \left (1+i {\mathrm e}^{i t}\right )-48 \operatorname {Catalan} \right ) \left (-\sec \left (t \right )+\tan \left (t \right )\right )}{24} \]

✓ Mathematica. Time used: 1.882 (sec). Leaf size: 813

ode=D[y[t],t]+y[t]*Sec[t]==t; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 9.074 (sec). Leaf size: 141

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + y(t)/cos(t) + Derivative(y(t), t),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\sqrt {\sin {\left (t \right )} - 1} \left (- \int \frac {t \sqrt {\sin {\left (t \right )} + 1}}{\sqrt {\sin {\left (t \right )} - 1}}\, dt + \int \limits ^{0} \frac {t \sqrt {\sin {\left (t \right )} + 1}}{\sqrt {\sin {\left (t \right )} - 1}}\, dt + \int \frac {\sqrt {\sin {\left (t \right )} + 1} y{\left (t \right )}}{\sqrt {\sin {\left (t \right )} - 1} \cos {\left (t \right )}}\, dt - \int \limits ^{0} \frac {\sqrt {\sin {\left (t \right )} + 1} y{\left (t \right )}}{\sqrt {\sin {\left (t \right )} - 1} \cos {\left (t \right )}}\, dt\right )}{\sqrt {\sin {\left (t \right )} - 1} \int \frac {\sqrt {\sin {\left (t \right )} + 1}}{\sqrt {\sin {\left (t \right )} - 1} \cos {\left (t \right )}}\, dt - \sqrt {\sin {\left (t \right )} + 1}} \]