74.2.16 problem 21

Internal problem ID [15857]
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 : Tuesday, January 28, 2025 at 08:10:23 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*}

✓ Solution by Maple

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

dsolve([diff(y(t),t$2)+4*y(t)=t,y(1/4*Pi) = 1, D(y)(1/4*Pi) = 1/16*Pi],y(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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