https://file.aiquickdraw.com/mathbot/file/17254439164373c6p1koe.jpg

The image you provided contains mathematical expressions. Let's go through each one in detail:

1. **First Expression:**
   \[
   y'(t - 2) + y(t) = 3
   \]

2. **Second Expression:**
   \[
   \text{The given expression is rewritten as:} \quad \text{From here, the expression is hard to read, but it seems to be stating a method to solve the equation or a specific manipulation of the equation.}
   \]

3. **Third Expression:**
   \[
   y(t) = e^{(t-2)} + (1-e^{-t}) (e^{(t+2)})
   \]

4. **Fourth Expression:**
   \[
   \text{Some expression seems to deal with finding the inverse or applying an inverse operation. The notation is somewhat ambiguous, but it might involve solving the equation.}
   \]

5. **Fifth Expression:**
   \[
   \int (t - 1) \delta(t-1) \, dt = 1
   \]

The last equation is an integral involving the Dirac delta function, which typically evaluates to 1 if the argument of the delta function matches the variable of integration at a specific point.

Would you like a detailed solution for any of these specific expressions, or do you have any other questions about the equations in the image?

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**5 Follow-Up Questions:**

1. Would you like to solve the first differential equation step by step?
2. Do you need a detailed explanation of the delta function integral?
3. Should we simplify the exponential expression in the third equation?
4. Do you need clarification on how to apply inverse operations in the context of these equations?
5. Would you like to know more about solving differential equations with shifting arguments like \( y(t - 2) \)?

**Tip:** When dealing with Dirac delta functions, remember that the integral of \( \delta(t-a) \) over a region including \( t=a \) is always 1.

Given the differential equation y'(t-2) + y(t) = 3, solve for y(t).

This page provides a detailed solution for the differential equation y'(t-2) + y(t) = 3. It includes explanations on the application of exponential functions and the Dirac delta function in solving differential equations. Suitable for undergraduate students.

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