To solve the partial differential equation \( \frac{\partial^2 y}{\partial x \partial t} = e^{-t} \) by direct integration, follow these steps:

### Step 1: Integrate with respect to \( x \)
First, we integrate the equation with respect to \( x \):

\[
\frac{\partial y}{\partial t} = \int e^{-t} \, dx
\]

Since \( e^{-t} \) is independent of \( x \), the integration gives:

\[
\frac{\partial y}{\partial t} = e^{-t}x + f(t)
\]

where \( f(t) \) is an arbitrary function of \( t \), arising from the integration.

### Step 2: Integrate with respect to \( t \)
Next, we integrate the result with respect to \( t \):

\[
y(x,t) = \int \left( e^{-t}x + f(t) \right) \, dt
\]

This gives:

\[
y(x,t) = -xe^{-t} + \int f(t) \, dt + g(x)
\]

where \( g(x) \) is an arbitrary function of \( x \), arising from the integration.

Thus, the general solution to the partial differential equation is:

\[
y(x,t) = -xe^{-t} + F(t) + g(x)
\]

where \( F(t) = \int f(t) \, dt \) is the indefinite integral of \( f(t) \).

Would you like more details on this process or have any questions? 

Here are some related questions you might consider:
1. What are the physical interpretations of arbitrary functions in differential equations?
2. How can boundary conditions be applied to find specific solutions?
3. What is the significance of the arbitrary function \( g(x) \)?
4. How does the method of separation of variables differ from direct integration?
5. How would this process change if the equation had different partial derivatives?

**Tip:** When solving partial differential equations, it's important to keep track of the variables with respect to which you are integrating, as arbitrary functions can emerge based on these variables.

Learn how to solve the partial differential equation ∂²y/∂x∂t = e^-t using direct integration. This problem involves techniques from partial differential equations and integration methods. Suitable for advanced undergraduate students and professionals.

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