40.6.14 problem 14

Internal problem ID [8646]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 6. Laplace Transforms. Problem set 6.2, page 216
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 05:40:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=50 t -100 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (2\right )&=-4 \\ y^{\prime }\left (2\right )&=14 \\ \end{align*}

✓ Maple. Time used: 0.102 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 50*t-100; 
ic:=[y(2) = -4, D(y)(2) = 14]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -24+2 \,{\mathrm e}^{-t +2} \sin \left (2 t -4\right )+10 t \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 25

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==50*t-100; 
ic={y[2]==-4,Derivative[1][y][2]==14}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 10 t-2 e^{2-t} \sin (4-2 t)-24 \end{align*}

✓ Sympy. Time used: 0.130 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-50*t + 5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) + 100,0) 
ics = {y(2): -4, Subs(Derivative(y(t), t), t, 2): 14} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 10 t + \left (\frac {2 e^{2} \sin {\left (2 t \right )} \cos {\left (4 \right )}}{\cos ^{2}{\left (4 \right )} + \sin ^{2}{\left (4 \right )}} - \frac {2 e^{2} \sin {\left (4 \right )} \cos {\left (2 t \right )}}{\cos ^{2}{\left (4 \right )} + \sin ^{2}{\left (4 \right )}}\right ) e^{- t} - 24 \]