87.22.36 problem 36

Internal problem ID [23781]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 36
Date solved : Thursday, October 02, 2025 at 09:45:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=13 t +17+40 \sin \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=30 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+13*y(t) = 13*t+17+40*sin(t); 
ic:=[y(0) = 30, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \left (30 \cos \left (3 t \right )+20 \sin \left (3 t \right )\right ) {\mathrm e}^{-2 t}+1+t -\cos \left (t \right )+3 \sin \left (t \right ) \]

✓ Mathematica. Time used: 0.19 (sec). Leaf size: 38

ode=D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==13*t+17+40*Sin[t]; 
ic={y[0]==30,Derivative[1][y][0] ==4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to t+3 \sin (t)+20 e^{-2 t} \sin (3 t)-\cos (t)+30 e^{-2 t} \cos (3 t)+1 \end{align*}

✓ Sympy. Time used: 0.181 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-13*t + 13*y(t) - 40*sin(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 17,0) 
ics = {y(0): 30, Subs(Derivative(y(t), t), t, 0): 4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = t + \left (20 \sin {\left (3 t \right )} + 30 \cos {\left (3 t \right )}\right ) e^{- 2 t} + 3 \sin {\left (t \right )} - \cos {\left (t \right )} + 1 \]