68.18.36 problem 42

Internal problem ID [17880]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 42
Date solved : Thursday, October 02, 2025 at 02:29:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)-4*y(t) = t; 
ic:=[y(0) = 2, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {17 \,{\mathrm e}^{2 t}}{16}+\frac {15 \,{\mathrm e}^{-2 t}}{16}-\frac {t}{4} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 16

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

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

✓ Sympy. Time used: 0.060 (sec). Leaf size: 22

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

\[ y{\left (t \right )} = - \frac {t}{4} + \frac {17 e^{2 t}}{16} + \frac {15 e^{- 2 t}}{16} \]