76.22.2 problem 15

Internal problem ID [17712]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.8 (Convolution Integrals and Their Applications). Problems at page 359
Problem number : 15
Date solved : Monday, March 31, 2025 at 04:25:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+25 y&=\sin \left (\alpha t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.115 (sec). Leaf size: 65

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+25*y(t) = sin(alpha*t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\alpha \left (24 \cos \left (4 t \right )+\sin \left (4 t \right ) \left (\alpha ^{2}-7\right )\right ) {\mathrm e}^{-3 t}-4 \alpha ^{2} \sin \left (\alpha t \right )-24 \alpha \cos \left (\alpha t \right )+100 \sin \left (\alpha t \right )}{4 \alpha ^{4}-56 \alpha ^{2}+2500} \]

✓ Mathematica. Time used: 1.535 (sec). Leaf size: 126

ode=D[y[t],{t,2}]+D[y[t],t]+25*y[t]==Sin[a*t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {e^{-t/2} \left (2 \sqrt {11} a^3 \sin \left (\frac {3 \sqrt {11} t}{2}\right )-33 a^2 e^{t/2} \sin (a t)-49 \sqrt {11} a \sin \left (\frac {3 \sqrt {11} t}{2}\right )+825 e^{t/2} \sin (a t)+33 a \cos \left (\frac {3 \sqrt {11} t}{2}\right )-33 a e^{t/2} \cos (a t)\right )}{33 \left (a^4-49 a^2+625\right )} \]

✓ Sympy. Time used: 0.365 (sec). Leaf size: 104

from sympy import * 
t = symbols("t") 
Alpha = symbols("Alpha") 
y = Function("y") 
ode = Eq(25*y(t) - sin(Alpha*t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {\mathrm {A}^{2} \sin {\left (\mathrm {A} t \right )}}{\mathrm {A}^{4} - 14 \mathrm {A}^{2} + 625} - \frac {6 \mathrm {A} \cos {\left (\mathrm {A} t \right )}}{\mathrm {A}^{4} - 14 \mathrm {A}^{2} + 625} + \left (\frac {6 \mathrm {A} \cos {\left (4 t \right )}}{\mathrm {A}^{4} - 14 \mathrm {A}^{2} + 625} + \frac {\left (\mathrm {A}^{3} - 7 \mathrm {A}\right ) \sin {\left (4 t \right )}}{4 \mathrm {A}^{4} - 56 \mathrm {A}^{2} + 2500}\right ) e^{- 3 t} + \frac {25 \sin {\left (\mathrm {A} t \right )}}{\mathrm {A}^{4} - 14 \mathrm {A}^{2} + 625} \]