The given differential equation is:

\[
\frac{dy}{dx} = x \cdot y
\]

This is a first-order separable differential equation. To solve it, we can separate the variables \( y \) and \( x \) and then integrate both sides. Here's the step-by-step solution:

### Step 1: Separate the variables
We rewrite the equation so that all terms involving \( y \) are on one side and all terms involving \( x \) are on the other side:

\[
\frac{1}{y} \, dy = x \, dx
\]

### Step 2: Integrate both sides
Now, we integrate both sides:

\[
\int \frac{1}{y} \, dy = \int x \, dx
\]

The integrals are:

\[
\ln |y| = \frac{x^2}{2} + C
\]

where \( C \) is the constant of integration.

### Step 3: Solve for \( y \)
To solve for \( y \), exponentiate both sides to eliminate the logarithm:

\[
|y| = e^{\frac{x^2}{2} + C}
\]

Since \( e^{C} \) is a constant, we can replace \( e^{C} \) with a new constant \( C_1 \), and we get:

\[
y = C_1 e^{\frac{x^2}{2}}
\]

Thus, the general solution to the differential equation is:

\[
y = C_1 e^{\frac{x^2}{2}}
\]

where \( C_1 \) is an arbitrary constant determined by initial conditions.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do initial conditions affect the value of \( C_1 \)?
2. Can this method be applied to other separable differential equations?
3. What is the significance of the constant \( C_1 \) in the solution?
4. How do you interpret the solution graphically for different values of \( C_1 \)?
5. How can this solution be used in real-world problems, like population growth?

**Tip:** When solving separable differential equations, always ensure the variables are correctly separated before integrating.

Explore the step-by-step solution to the separable differential equation dy/dx = xy. Learn how to separate variables, integrate both sides, and find the general solution. Suitable for advanced high school and college-level students studying calculus.

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