91Ó°ÊÓ

A separable differential equation is a type of differential equation that allows you to separate the variables involved. By separating variables, we can isolate each variable on different sides of the equation. This facilitates the integration of each side. In essence, a differential equation is separable if it can be rewritten in the form:

• \( \frac{dy}{dx} = g(x)h(y) \)

This property enables us to manipulate and solve for \( y \) by using standard integration techniques after separating the variables:

• \( \int \frac{1}{h(y)} \, dy = \int g(x) \, dx \)

In the original exercise, the differential equation \( \frac{dy}{dx} = xy \) is indeed separable. It can be converted by dividing both sides by \( y \) and multiplying by \( dx \) to yield:

• \( \frac{1}{y} \, dy = x \, dx \)

With the separated equation, solving it involves integrating both sides to find the function \( y(x) \) which describes the solution of the curve.

Initial conditions are specific values provided to determine the constants after integrating a differential equation. They are given as starting points, or pre-determined values, that guarantee a specific solution curve among the many possible solutions to a differential equation. These conditions are crucial to ensure the correct mathematical modeling of physical scenarios or applications related to the problem.In the context of the original exercise, the initial condition provided is the point \((0, 1)\). This information is used to find the constant \( C \) in the integrated function. After solving the separated differential equation, we ended up with the general solution of the form:

• \( y = Ce^{\frac{x^2}{2}} \)

Applying the initial condition \( x = 0 \) and \( y = 1 \) helps us find the value of \( C \), which is confirmed as \( 1 \):

• \( 1 = C \cdot e^0 \Rightarrow C = 1 \)

So, the specific solution passing through \((0, 1)\) is determined as \( y = e^{\frac{x^2}{2}} \).

The slope of a curve at any point is represented as the derivative \( \frac{dy}{dx} \). It visually describes how the curve is 'climbing' or 'falling' at that particular point, relating to the steepness. In calculus, the slope gives insight into the behavior of functions and their rates of change.For this particular problem, we're given that the slope at any point \((x, y)\) is \( xy \). This defines a unique modeling of the differential equation that describes how steeply the curve moves based on both the \( x \) and \( y \) values. This relationship sets up the base for the entire differential equation that we solved:

• \( \frac{dy}{dx} = xy \) - this equation articulates the relationship between the slope of the curve and the coordinates \( x \) and \( y \).

The solution curve finds its form according to both this slope condition and the initial conditions. By checking that the derivative of our solution, \( y = e^{\frac{x^2}{2}} \), follows this specific slope formula, we've verified that it fulfills the conditions provided by the original problem.