Solving Differnetial Equations Using Laplace Transform Find the solution of the second-order differential equation, assuming all initial conditions are zeros: x″(t) + 5x′(t) + 6x(t) - 50 = 0

Question

Solving Differnetial Equations Using Laplace Transform

Find the solution of the second-order differential equation, assuming all initial conditions are zeros:

x″(t) + 5x′(t) + 6x(t) - 50 = 0

Accepted Answer

Applying Laplace transform:

[latex]s^2 X(s)-s x(0)-x^{\prime}(0)+5 s X(s)-5 x(0)+6 X(s)-\frac{50}{s}=0[/latex]

Setting x(0) = x′(0) = 0 :

[latex]\begin{gathered}\left(s^2+5 s+6 ight) X(s)=\frac{50}{s} \X(s)=\frac{50}{s\left(s^2+5 s+6 ight)}=\frac{50}{s(s+2)(s+3)}=\frac{8.33}{s}-\frac{25}{s+2}+\frac{16.67}{s+3}\end{gathered}[/latex]

Using Table B.1 to find inverse Laplace transform results in:

[latex]x(t)=8.33-25 e^{-2 t}+16.67 e^{-3 t}, \quad t>0[/latex]

TABLE B.1 The Laplace Transform of Basic Functions and Some of its Properties
x(t)	X(s)
Basic Functions
δ(t)	1
1	[latex]\frac{1}{s}[/latex]
t	[latex]\frac{1}{s^2}[/latex]
[latex]t^n[/latex]	[latex]\frac{n !}{s^{n+1}}[/latex]
[latex]e^{at}[/latex]	[latex]\frac{1}{s-a}[/latex]
cos(ωt)	[latex]\frac{s}{s^2+\omega^2}[/latex]
sin(ωt)	[latex]\frac{\omega}{s^2+\omega^2}[/latex]
s-domain shift
[latex]x(t)e^{at}[/latex]	[latex]X(s-a)[/latex]
[latex]te^{at}[/latex]	[latex]\frac{1}{(s-a)^2}[/latex]
[latex]\cos (\omega t)e^{at}[/latex]	[latex]\frac{s-a}{(s-a)^2+\omega^2}[/latex]
[latex]\sin (\omega t)e^{at}[/latex]	[latex]\frac{\omega}{(s-a)^2+\omega^2}[/latex]
t-domain shift
x(t - a)	[latex]X(s) e^{-a s}[/latex]
Differentiation
x′(t)	sX(s) - x(0)
x″(t)	s²X(s) - sx(0) - x′(0)
[latex]x^{(n)}(t)[/latex]	[latex]s^n X(s)-s^{n-1} x(0)-s^{n-2} x^{\prime}(0)-\cdots-x^{(n-1)}(0)[/latex]
Integration
[latex]\int_0^t x( au) d au[/latex]	[latex]\frac{1}{s} X(s)[/latex]
Convolution
[latex]x_1(t) * x_2(t)[/latex]	[latex]X_1(s) \cdot X_2(s)[/latex]
[latex]x_1(t) \cdot x_2(t)[/latex]	[latex]X_1(s) * X_2(s)[/latex]

Question B.6: Solving Differnetial Equations Using Laplace Transform Find ...

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