61.4.2 problem Problem 2(b)

Internal problem ID [15327]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(b)
Date solved : Thursday, October 02, 2025 at 10:11:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{\prime \prime }+16 y^{\prime }+17 y&=17 t -1 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.094 (sec). Leaf size: 17

ode:=4*diff(diff(y(t),t),t)+16*diff(y(t),t)+17*y(t) = 17*t-1; 
ic:=[y(0) = -1, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 21

ode=4*D[y[t],{t,2}]+16*D[y[t],t]+17*y[t]==17*t-1; 
ic={y[0]==-1,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.133 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-17*t + 17*y(t) + 16*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)) + 1,0) 
ics = {y(0): -1, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

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