Differentials

What are differentials?

Finding the inverse of an implicit derivative. For example, the implicit derivative could be:

\[ \frac{dy}{dx} = 5x^2 -2x \]

When solving for a differential, there are general solutions and specific solutions, where the concept is similar to indefinite integrals and definite integrals.

Orders

Orders of differentials refer to the degree of the derivative. For example, the equation above is a first order differential. A possible second order differential would then be

\[ \frac{d^2y}{dx^2} = 5x^2 -2x \]

Homogenous and non-homogenous

Separable equations

Separable equations are easier to solve. However, they must be of a certain pattern:

\[ \frac{dy}{dx} = f(x) \times g(x) \]

or

\[ \frac{dy}{dx} = \frac{f(x)}{g(x)} \]

To solve separable equations, first balance out the equation to look like:

\[ \int{f(x)}{dx} = \int{g(x)}{dy} \]

Then, solve the integral.

Non separable equations

There are times when equation are not separable. For example, the following equation is not separable:

\[ \frac{dy}{dx} + P(x)y = Q(x) \]

We can solve the non-separable equations using this thing called an integrating factor.

\[ I = e^{\int{P(X)}{dx}} \]

After finding the integrating factor, place it in the equation to solve for the general solution:

\[ Iy = \int{IQ(X)}{dx} \]

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Last update: June 11, 2023
Created: June 11, 2023