58.20.11 problem 11

Internal problem ID [14935]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 11
Date solved : Thursday, October 02, 2025 at 09:56:42 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-5 y^{\prime \prime }+7 y^{\prime }-3 y&=20 \sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.187 (sec). Leaf size: 19

ode:=diff(diff(diff(y(t),t),t),t)-5*diff(diff(y(t),t),t)+7*diff(y(t),t)-3*y(t) = 20*sin(t); 
ic:=[y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = -2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (3-4 t \right ) {\mathrm e}^{t}-3 \cos \left (t \right )+\sin \left (t \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 21

ode=D[y[t],{t,3}]-5*D[y[t],{t,2}]+7*D[y[t],t]-3*y[t]==20*Sin[t]; 
ic={y[0]==0,Derivative[1][y][0]==0,Derivative[2][y][0]==-2}; 
DSolve[{ode,ic},{y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^t (3-4 t)+\sin (t)-3 \cos (t) \end{align*}

✓ Sympy. Time used: 0.163 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*y(t) - 20*sin(t) + 7*Derivative(y(t), t) - 5*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (3 - 4 t\right ) e^{t} + \sin {\left (t \right )} - 3 \cos {\left (t \right )} \]