70.22.6 problem 19

Internal problem ID [19073]
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 : 19
Date solved : Thursday, October 02, 2025 at 03:37:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\cos \left (\alpha t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.191 (sec). Leaf size: 76

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+2*y(t) = cos(alpha*t); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {\left (-\alpha ^{4}-3 \alpha ^{2}-2\right ) {\mathrm e}^{-2 t}+\left (2 \alpha ^{4}+9 \alpha ^{2}+4\right ) {\mathrm e}^{-t}-\cos \left (\alpha t \right ) \alpha ^{2}+3 \alpha \sin \left (\alpha t \right )+2 \cos \left (\alpha t \right )}{\left (\alpha ^{2}+1\right ) \left (\alpha ^{2}+4\right )} \]

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 103

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

\begin{align*} y(t)&\to e^{-2 t} \left (\int _1^t-e^{2 K[1]} \cos (a K[1])dK[1]-e^t \int _1^0e^{K[2]} \cos (a K[2])dK[2]+e^t \int _1^te^{K[2]} \cos (a K[2])dK[2]-\int _1^0-e^{2 K[1]} \cos (a K[1])dK[1]+2 e^t-1\right ) \end{align*}

✓ Sympy. Time used: 0.198 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
Alpha = symbols("Alpha") 
y = Function("y") 
ode = Eq(2*y(t) - cos(Alpha*t) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {\mathrm {A}^{2} \cos {\left (\mathrm {A} t \right )}}{\mathrm {A}^{4} + 5 \mathrm {A}^{2} + 4} + \frac {3 \mathrm {A} \sin {\left (\mathrm {A} t \right )}}{\mathrm {A}^{4} + 5 \mathrm {A}^{2} + 4} + \frac {\left (- \mathrm {A}^{2} - 2\right ) e^{- 2 t}}{\mathrm {A}^{2} + 4} + \frac {2 \cos {\left (\mathrm {A} t \right )}}{\mathrm {A}^{4} + 5 \mathrm {A}^{2} + 4} + \frac {\left (2 \mathrm {A}^{2} + 1\right ) e^{- t}}{\mathrm {A}^{2} + 1} \]