# Solve Dirac Delta Function IVP: y''-2y'-3y=2\delta (t-1)-\delta (t-3)

• Precursor
In summary, the given symbolic initial value problem is solved using the laplace transform. The final answer is y(t)=e^{1-t}(e^{4t-4}-1)u(t-1)-2e^{3-t}(e^{4t-12}-1)u(t-3)+e^{-t}+e^{3t}.
Precursor
Homework Statement
Solve the given symbolic initial value problem: $$y''-2y'-3y=2\delta (t-1)-\delta (t-3) ;y(0)=2,y'(0)=2$$

The attempt at a solution

Let Y(s):= L{y(t)}(s)

Taking laplace transform of both sides:

$$[s^{2}Y(s)-2s-2]-2[sY(s)-2]-3Y(s)=2e^{-s}-e^{-3s}$$
$$s^{2}Y(s)-2sY(s)-3Y(s)=2e^{-s}-e^{-3s}+2s-2$$
$$Y(s)=\frac{2e^{-s}-e^{-3s}+2s-2}{s^{2}-2s-3}$$
$$Y(s)=\frac{2e^{-s}}{s^{2}-2s-3}- \frac{e^{-3s}}{s^{2}-2s-3}+\frac{2s-2}{s^{2}-2s-3}$$
$$y(t)=e^{-(t-1)}\frac{1}{2}e^{-(t-1)}(e^{4(t-1)}-1)u(t-1)-e^{-(t-3)}e^{-(t-3)}(e^{4(t-3)}-1)u(t-3)+e^{-t}+e^{3t}$$

And my final answer:

$$y(t)=e^{1-t}(e^{4t-4}-1)u(t-1)-2e^{3-t}(e^{4t-12}-1)u(t-3)+e^{-t}+e^{3t}$$

Is this correct?

It looks to be right sort of thing to be doing.

## 1. What is the Dirac Delta function?

The Dirac Delta function, denoted as δ(t), is a mathematical concept used in engineering and physics to represent an impulse or spike function. It is defined as infinite at t = 0 and zero everywhere else, with an area of 1 under the curve. It is often used to model an instantaneous change or point source in a system.

## 2. What does the term "IVP" mean in this context?

"IVP" stands for Initial Value Problem, which is a type of mathematical problem that involves finding the solution to a differential equation with given initial conditions. In the context of this question, it means we are trying to solve the differential equation with the given initial conditions at t = 0.

## 3. What is the significance of the constants 2 and 3 in the differential equation?

The constants 2 and 3 represent the coefficients of the first and second derivatives of the unknown function y. They determine the behavior of the solution and can be used to classify the type of differential equation (e.g. linear or non-linear, homogeneous or non-homogeneous).

## 4. How do you solve a differential equation with Dirac Delta functions?

When solving a differential equation with Dirac Delta functions, we first need to find the general solution to the homogeneous part of the equation (i.e. without the Dirac Delta functions). Then, we use the initial conditions to find the particular solution to the non-homogeneous part (i.e. the Dirac Delta functions). Finally, we combine the two solutions to get the overall solution to the differential equation.

## 5. How do you interpret the solution to this particular differential equation?

The solution to this differential equation represents the behavior of a system with two point sources of impulses (spikes) at t = 1 and t = 3. The function y(t) represents the response of the system to these impulses, and the solution will be different depending on the initial conditions given.

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