Differential equations 

Differential equations help us to understand phenomena that involve rates of change. [ENGINE BEING REVVED]   For example, differential equations help us understand the spread of disease, weather and climate prediction, traffic flow, financial markets, population growth, water pollution, chemical reactions, suspension bridges, brain function, rockets, tumor growth, radioactive decay airflow across a plane's wing, electrical circuits, planetary motion, and the vibration of guitar strings.     OK,

but what is a differential equation?   From calculus, we know the word differentiate means to compute a derivative, and one definition of a differential equation is an equation containing a derivative,   and this actually explains to some extent the wide applicability of differential equations because remember that a derivative represents a rate of change, and that rate of change can be anything from a rate of population growth to a rate of radioactive decay.   Let's take a look at some examples. Here's one. The derivative of y with respect to x plus xy equals e to the 2x. That's an equation containing a derivative, so it's a differential equation. 

  Oh, and remember that dy/dx and y prime are two ways to write the derivative.   Using the prime notation, here's an example involving a second-order derivative. y double prime minus y prime plus y squared equals sine of x If the equation contains at least one derivative of any order, then it's a differential equation.  

 Let's consider one more example.   If you've taken multivariable calculus, then you'll have seen partial derivatives, and these come up in differential equations too.   Here's a somewhat famous example: u xx plus u yy equals zero.   Alright, now that you know what a differential equation is, the next step is to solve one. In the next video, we'll solve a simple differential equation, and we'll discuss how it's different from familiar algebraic equations.   See you then! 

differential equation: how to check a differential equation is correct solution

Q.              y''+2y'+25y=10 cos(5x)

sol....

   Okay, let's say you have the equation y double prime plus 2 y prime plus 25 y equals 10 cosines of 5 x, and you think you have a solution.   How do you check if yours is a correct solution?  To find out how to let's check that y equals sine of five x is a solution to our equation.  Remember that y is a solution if it makes a true equation when we plug it in, and I like to start by plugging y into the left-hand side, which I've abbreviated as L H S.  When we do that, we get a sign of five x double prime plus two sines of five x prime plus 25 sines of five x.

  Now we need to know what the first and second derivatives are, so let's calculate those separately.  To find the derivative of sine of five x, we just remember that the derivative of sine is cosine, so we get the cosine of five x, and then we have to use the chain rule, so we'll multiply by the derivative of the inner function, which is five.  Alright. To find the second derivative, we just take the derivative of the first derivative, and we get five times the derivative of the cosine of five x, and we know the derivative of cosine is negative sine, so we'll get negative sine of five x, and again we multiply by the derivative of the inner function, which is five, so we get negative 25 sines of five x.  Now we can rewrite the left-hand side of our equation by replacing the second derivative with a negative 25 sign of five x, and the first derivative with five cosines of five x.

  To simplify we see that negative 25 sign of five x and 25 sines of five x cancel, and we're left with the middle term, which is 10 cosine of five x.  Then we compare that to the right-hand side of the equation, and that's just 10 cosines of five x. We don't have to plug in anything this time. Since the left-hand side and right-hand side are in fact equal, y equals sine of five x is a solution, and we're finished. Now that this quick exercise has clarified what it means to say a function is a solution to a differential equation,  we'll move on to a new topic in the next video, as promised. See you then!