74.2.16 problem 21

Internal problem ID [15778]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Review exercises, page 23
Problem number : 21
Date solved : Thursday, March 13, 2025 at 06:19:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=t \end{align*}

With initial conditions

\begin{align*} y \left (\frac {\pi }{4}\right )&=1\\ y^{\prime }\left (\frac {\pi }{4}\right )&=\frac {\pi }{16} \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 30

ode:=diff(diff(y(t),t),t)+4*y(t) = t; 
ic:=y(1/4*Pi) = 1, D(y)(1/4*Pi) = 1/16*Pi; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {\cos \left (2 t \right ) \left (-\pi +4\right )}{32}+\frac {\left (-2 \pi +32\right ) \sin \left (2 t \right )}{32}+\frac {t}{4} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 31

ode=D[y[t],{t,2}]+4*y[t]==t; 
ic={y[Pi/4]==1,Derivative[1][y][Pi/4]==Pi/16}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{32} (8 t-2 (\pi -16) \sin (2 t)-(\pi -4) \cos (2 t)) \]

✓ Sympy. Time used: 0.103 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(pi/4): 1, Subs(Derivative(y(t), t), t, pi/4): pi/16} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t}{4} + \left (1 - \frac {\pi }{16}\right ) \sin {\left (2 t \right )} + \left (\frac {1}{8} - \frac {\pi }{32}\right ) \cos {\left (2 t \right )} \]