Question

Solve each differential equation. $$d y=x^{2} d x$$

Short answer

The solution of the differential equation is \(y = \frac{x^3}{3} + C\).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is a separable equation, which means that it can be written in the form of a product of a function of y and a function of x, and these can be separated on different sides of the equation.

Separate Variables

The variables can be separated by moving all terms involving y to one side of the equation and all terms involving x to the other side. In this case, we can write it as \(dy = x^2 dx\). The variables are already separated.

Integrate Both Sides

Since the variables are separated, integrate both sides of the equation. The integral of \(dy\) is \(y\), and the integral of \(x^2 dx\) is \(\frac{x^3}{3}\). So the integrals are $$\int dy = \int x^2 dx$$ which results in $$y = \frac{x^3}{3} + C$$ where C is the constant of integration.

Key concepts

Separable Differential Equations

Separable differential equations are a specific type of first-order differential equation where you can separate variables into two sides of the equation: one side containing only x-terms and the other containing only y-terms. This separation allows us to integrate each side with respect to its respective variable.
Consider the problem of solving a differential equation such as \(dy = x^2 dx\). To solve this, first recognize that the terms involving y and the terms involving x are already separated. With this type of equation, our strategy involves integrating each side independently to find the relation between the variables. In the given exercise step where we integrate \(dy\) on one side and \(x^2 dx\) on the other, we are taking advantage of the separable nature of the equation.

Integrating Factors

An integrating factor is a function we multiply by a differential equation to facilitate its solution, usually transforming a non-separable equation into a separable one. While not explicitly needed for the immediately separable equations, integrating factors are incredibly useful for more complex differential equations that are not easily separated.
For example, if we had an equation where separation is not straightforward, we would look for an integrating factor that simplifies the equation. This process involves identifying a function that, when multiplied with the original differential equation, allows the terms to be organized into a recognizable pattern that can be integrated.

Constant of Integration

The constant of integration appears when we evaluate an indefinite integral. Since antiderivatives are not unique (adding any constant results in another valid antiderivative), we include the constant of integration, typically denoted as \(C\), to represent all possible antiderivatives.
When we integrated \(dy\) and got \(y\), and integrated \(x^2 dx\) which resulted in \(\frac{x^3}{3}\), the addition of \(C\) to one side is essential to cover all antiderivatives. When solving a differential equation, it's crucial to remember that this constant represents the family of all possible functions that could be solutions to the differential equation, given the general nature of antidifferentiation.