Linear Differential Equations

Math Easy Solutions

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In this video I go over the derivation of the solution to a first-order linear differential equation. This type of equation involves a function's first derivative (hence "first-order") and the function in the form: y' + P(x)y = Q(x). In other words, the function, y, appears in a linear form with no higher power being present, such as y^2, y^3, etc. This type of equation is not separable thus the methods in my earlier video will not work in solving these types of equations. That being said, we can still solve this equation by first making it in the form of a separable equation. This is done by multiplying both sides of the equation by an "integrating factor" I(x) such that the derivative of the combination of [I(x)y] is equal to I(x)[y' + P(x)y]. It turns out that every first order linear differential equation can be manipulated in such a way.

From this equation manipulation, I show how to derive the general solution to the equation, as well as solving for the integrating factor I(x), which also involves obtaining another separable equation. Although the resulting solution is very large, it is easiest not to memorize the solution but rather remember the form of the integrating factor, which we multiple both sides of the linear equation and then integrate both sides.

In the derivation, I also go over a simple example in solving for the solution of the equation: xy' + y = 2x. This is a pretty extensive derivation on linear differential equation so make sure to watch it as these types of equation arise many times in science and physics.

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhtp1wmzxIfxq33uuWg

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/linear-differential-equations