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.