PLEASE READ THIS FIRST: Okay I am not interested in the solution to the differnt

Question

PLEASE READ THIS FIRST:

Okay I am not interested in the solution to the differntial equation, I am not  interested in the inverse

Laplace of the problem and I am not interested in the Laplace transform of y''+4y'+14 section of thestyle="font-size: 13px;" data-mce-style="font-size: 13px;">

equation, I am only interested in the setting up T(t) as a heaviside equation and taking the Laplacestyle="font-size: 13px;" data-mce-style="font-size: 13px;">style="text-decoration: underline;" data-mce-style="text-decoration: underline;">style="font-size: 13px;" data-mce-style="font-size: 13px;">

transform of that and please explain in detail!style="font-size: 13px;" data-mce-style="font-size: 13px;">style="font-size: 13px;" data-mce-style="font-size: 13px;">style="text-decoration: underline;" data-mce-style="text-decoration: underline;">style="font-size: 13px;" data-mce-style="font-size: 13px;">

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Here is what I have so far for the heavisde function:

                                        t(1-step(t-(1/2)))+(1-t)(step(t-(1/2))-step(t-1))

Now when I graph it on wolfram it gives me the right graph

          http://www.wolframalpha.com/input/?i=plot++t(1-step(t-.5))%2b(1-t)(step(t-.5)-step(t-1))

but it appears as though the graph should be repeated infinitley which is why they give T(t+t)=T(t) but I

don't understand how to incorporate that as part of the heaviside equation. Please explain how thoroughly>style="font-size: 13px;">

I would like to understand well. The final solution is the Laplace transform of the given equationstyle="font-size: 13px;">style="font-size: 13px;">

(y''+4y'+14=T(t)) but please focus on the T(t) part.>style="font-size: 13px;">

Y(s) = L{T(t)} / (s^2+ 4s+14)

heaviside T(t) =

L{T(t)} =

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HERE IS THE ACTUAL PROBLEM:

                                           http://i.imgur.com/IvKsrEu.png

Thanks in advanced! :)

(Sorry for poor formating, I took a bunch of time to make it look neat and added pictures but

chegg screwed it all up and this is the best I could get to show up).

Okay I am not interested in the solution to the differntial equation, I am not interested in the inverse Laplace of the problem and I am not interested in the Laplace transform of y''+4y'+14 section of the equation, I am only interested in the setting up T(t) as a heaviside equation and taking the Laplace transform of that and please explain in detail! Here is what I have so far for the heavisde function: t(1-step(t-(1/2)))+(1-t)(step(t-(1/2))-step(t-1)) Now when I graph it on wolfram it gives me the right graph

Explanation / Answer

Solve 4 ( dy(x))/( dx)+( d^2 y(x))/( dx^2)+14 = 0: The general solution will be the sum of the complementary solution and particular solution. Find the complementary solution by solving ( d^2 y(x))/( dx^2)+4 ( dy(x))/( dx) = 0: Assume a solution will be proportional to e^(lambda x) for some constant lambda. Substitute y(x) = e^(lambda x) into the differential equation: ( d^2 )/( dx^2)(e^(lambda x))+4 ( d)/( dx)(e^(lambda x)) = 0 Substitute ( d^2 )/( dx^2)(e^(lambda x)) = lambda^2 e^(lambda x) and ( d)/( dx)(e^(lambda x)) = lambda e^(lambda x): lambda^2 e^(lambda x)+4 lambda e^(lambda x) = 0 Factor out e^(lambda x): (lambda^2+4 lambda) e^(lambda x) = 0 Since e^(lambda x) !=0 for any finite lambda, the zeros must come from the polynomial: lambda^2+4 lambda = 0 Factor: lambda (lambda+4) = 0 Solve for lambda: lambda = -4 or lambda = 0 The root lambda = -4 gives y_1(x) = c_1 e^(-4 x) as a solution, where c_1 is an arbitrary constant. The root lambda = 0 gives y_2(x) = c_2 as a solution, where c_2 is an arbitrary constant. The general solution is the sum of the above solutions: y(x) = y_1(x)+y_2(x) = c_1/e^(4 x)+c_2 Determine the particular solution to ( d^2 y(x))/( dx^2)+4 ( dy(x))/( dx) = -14 by the method of undetermined coefficients: The particular solution to ( d^2 y(x))/( dx^2)+4 ( dy(x))/( dx) = -14 is of the form: y_p(x) = a_1 x, where a_1 was multiplied by x to account for 1 in the complementary solution. Solve for the unknown constant a_1: Compute ( dy_p(x))/( dx): ( dy_p(x))/( dx) = ( d)/( dx)(a_1 x) = a_1 Compute ( d^2 y_p(x))/( dx^2): ( d^2 y_p(x))/( dx^2) = ( d^2 )/( dx^2)(a_1 x) = 0 Substitute the particular solution y_p(x) into the differential equation: ( d^2 y_p(x))/( dx^2)+4 ( dy_p(x))/( dx) = -14 4 a_1 = -14 Solve the equation: a_1 = -7/2 Substitute a_1 into y_p(x) = a_1 x: y_p(x) = -(7 x)/2 The general solution is: Answer: | | y(x) = y_c(x)+y_p(x) = -(7 x)/2+c_1/e^(4 x)+c_2

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