68.12.43 problem 43

Internal problem ID [17637]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 43
Date solved : Thursday, October 02, 2025 at 02:26:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\tan \left (t \right )^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+y(t) = tan(t)^2; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \sin \left (t \right )+2 \cos \left (t \right )-2+\sin \left (t \right ) \ln \left (\sec \left (t \right )+\tan \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 59

ode=D[y[t],{t,2}]+y[t]==Tan[t]^2; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\cos (t) \int _1^0-\sin (K[1]) \tan ^2(K[1])dK[1]+\cos (t) \int _1^t-\sin (K[1]) \tan ^2(K[1])dK[1]+\sin (t) (\text {arctanh}(\sin (t))-\sin (t)+1) \end{align*}

✓ Sympy. Time used: 0.228 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - tan(t)**2 + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {\log {\left (\sin {\left (t \right )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\left (t \right )} + 1 \right )}}{2} + 1 + \frac {i \pi }{2}\right ) \sin {\left (t \right )} + 2 \cos {\left (t \right )} - 2 \]