90.14.19 problem 28

Internal problem ID [25247]
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 : 28
Date solved : Thursday, October 02, 2025 at 11:59:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y&=2-8 t \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 19

ode:=diff(diff(y(t),t),t)-4*y(t) = -8*t+2; 
ic:=[y(0) = 0, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{-2 t}}{2}+{\mathrm e}^{2 t}+2 t -\frac {1}{2} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 26

ode=D[y[t],{t,2}]-4*y[t]==2-8*t; 
ic={y[0]==0,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 t-\frac {e^{-2 t}}{2}+e^{2 t}-\frac {1}{2} \end{align*}

✓ Sympy. Time used: 0.078 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(8*t - 4*y(t) + Derivative(y(t), (t, 2)) - 2,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 2 t + e^{2 t} - \frac {1}{2} - \frac {e^{- 2 t}}{2} \]