68.11.36 problem 48

Internal problem ID [17573]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 48
Date solved : Thursday, October 02, 2025 at 02:25:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y&=32 t \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 14

ode:=diff(diff(y(t),t),t)-4*y(t) = 32*t; 
ic:=[y(0) = 0, D(y)(0) = 6]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -8 t +7 \sinh \left (2 t \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 27

ode=D[y[t],{t,2}]-4*y[t]==32*t; 
ic={y[0]==0,Derivative[1][y][0] ==6}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -8 t-\frac {7 e^{-2 t}}{2}+\frac {7 e^{2 t}}{2} \end{align*}

✓ Sympy. Time used: 0.055 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-32*t - 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 6} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - 8 t + \frac {7 e^{2 t}}{2} - \frac {7 e^{- 2 t}}{2} \]