https://file.aiquickdraw.com/mathbot/file/1725303122514fv3er9xj.jpg

Let's examine the second problem from the image.

### The Problem:
\[ \frac{dy}{dt} = y e^{t} - 2y + e^t + 7 \]
with the initial condition \( y(0) = 5 \).

### Steps to Solve:
This is a first-order linear differential equation. We can solve it using the method of integrating factors.

1. **Rewrite the Equation:**
   \[
   \frac{dy}{dt} - (e^t - 2)y = e^t + 7
   \]
   Here, the equation is in the standard form:
   \[
   \frac{dy}{dt} + P(t)y = Q(t)
   \]
   where \( P(t) = -(e^t - 2) \) and \( Q(t) = e^t + 7 \).

2. **Find the Integrating Factor:**
   The integrating factor \( \mu(t) \) is given by:
   \[
   \mu(t) = e^{\int P(t) \, dt}
   \]
   Let's calculate this:
   \[
   \mu(t) = e^{\int -(e^t - 2) \, dt} = e^{- \int (e^t - 2) \, dt}
   \]
   Now, compute the integral:
   \[
   \int (e^t - 2) \, dt = e^t - 2t
   \]
   So, the integrating factor becomes:
   \[
   \mu(t) = e^{-(e^t - 2t)} = e^{-e^t + 2t}
   \]

3. **Multiply the Differential Equation by the Integrating Factor:**
   Multiply the entire differential equation by \( \mu(t) = e^{-e^t + 2t} \):
   \[
   e^{-e^t + 2t} \frac{dy}{dt} - (e^t - 2)e^{-e^t + 2t} y = (e^t + 7)e^{-e^t + 2t}
   \]

   The left-hand side becomes the derivative of \( y \mu(t) \):
   \[
   \frac{d}{dt} \left( y e^{-e^t + 2t} \right) = (e^t + 7)e^{-e^t + 2t}
   \]

4. **Integrate Both Sides:**
   Integrate both sides with respect to \( t \):
   \[
   y e^{-e^t + 2t} = \int (e^t + 7)e^{-e^t + 2t} \, dt
   \]

   This integral can be solved by splitting it:
   \[
   \int e^t e^{-e^t + 2t} \, dt + 7 \int e^{-e^t + 2t} \, dt
   \]

   Both integrals can be challenging and typically require substitution methods or special functions, but let's assume we're looking for a solution in a simplified form or numerically.

5. **Apply the Initial Condition:**
   Substitute \( y(0) = 5 \) into the general solution to find the specific constant for the particular solution.

Given that the integrals are complex, the solution might be expressed implicitly. If an exact solution is required, more advanced techniques or computational tools could be employed.

Would you like to proceed with finding an exact solution, discuss further steps, or have any other questions?

Learn how to solve the first-order linear differential equation \( \frac{dy}{dt} = y e^{t} - 2y + e^t + 7 \) with the initial condition \( y(0) = 5 \). This problem involves using integrating factors and solving for a particular solution. Suitable for advanced undergraduate mathematics students.

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