90.14.7 problem 16

Internal problem ID [25235]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 3. Second Order Constant Coefficient Linear Differential Equations. Exercises at page 243
Problem number : 16
Date solved : Thursday, October 02, 2025 at 11:59:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=1+{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

ode:=diff(diff(y(t),t),t)+4*y(t) = 1+exp(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 +\frac {1}{4}+\frac {{\mathrm e}^{t}}{5}+\frac {\cos \left (2 t \right )}{4} \]

✓ Mathematica. Time used: 0.094 (sec). Leaf size: 34

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

\begin{align*} y(t)&\to \frac {e^t}{5}+\left (\frac {1}{4}+c_1\right ) \cos (2 t)+c_2 \sin (2 t)+\frac {1}{4} \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )} + \frac {e^{t}}{5} + \frac {1}{4} \]