Let α, β∈ℝ. Suppose f, g: ℝ →ℝ are differentiable functions such that f^'=αf+βg, g^'=αg+βf,

step 1 So here's a nice little proof with derivatives. So we're given some that A is bigger than one, e

Let α, β∈ℝ. Suppose f, g: ℝ →ℝ are differentiable functions...

Key Concepts

Hyperbolic Functions

Hyperbolic functions, including the hyperbolic sine and cosine, are analogs of the trigonometric functions but are defined using exponential functions. They naturally appear in the solutions of certain differential equations and have properties that mirror those of exponential growth and decay.

Exponential Functions

Exponential functions are a fundamental class of functions often encountered as solutions to differential equations where the rate of change of a function is proportional to the function itself. They serve as building blocks for constructing more complex solutions.

Linear Differential Equations with Constant Coefficients

These are equations where derivatives of the unknown function are multiplied by constant coefficients. The linearity and constant coefficients often lead to solutions expressible in terms of exponential functions, and the associated characteristic equation helps determine the form of the solution.

Systems of Differential Equations

This concept involves multiple interrelated differential equations that must be solved simultaneously. Techniques such as decoupling or diagonalization are used to reduce the system to simpler equations, making it easier to solve for each function individually.

Initial Value Problems

These problems specify the values of the unknown function(s) at a particular point, ensuring a unique solution by determining any arbitrary constants that arise in the general solution of a differential equation.

Transcript

00:02 So here's a nice little proof with derivatives.

00:05 So we're given some that a is bigger than one, exclusively bigger than one, and are strictly greater than one.

00:15 And then the absolute value of f is less than or equal to absolute value with x to the power of a.

00:22 And we're going to show that f is differentiable at zero.

00:25 So for it to exist or for it to be differentiable, well first off, the function needs to exist.

00:32 There at the point and also the limit at that needs at the point needs to exist.

00:41 So what does that mean? let's deal with the first one.

00:44 Let's deal with as to exist at the point.

00:47 So we don't we want f of x of zero to be some number.

00:54 So let's let's go ahead and do that.

00:58 So it can be not a number.

01:00 It can be like infinity or anything.

01:03 Or it doesn't work.

01:05 It has to have a value.

01:06 So that's where this comes into play.

01:09 The absolute value of f of x being less than or equal to, boy, this is, so this function is always less than or equal to the absolute value of a.

01:32 And just think about this, or a value of x to the eighth power.

01:37 Well, this is going to be, if you think of what this is, what this graph is going to look like, like this, and this, and absolute value is going to be, to have this kind of open up like this the whole time.

02:01 It's never going to go do anything else.

02:04 This is what's going to be.

02:05 And it's always positive.

02:07 And so the absolute value of this thing is always less than equal to it.

02:10 So if we imagine f zooming along here doing what it's going to do, at x is zero, you cannot be non -negative.

02:20 And it's always going to be less than an equal to that.

02:23 And when this function, when the absolute value of x to the eighth power, when you have a zero in there, well, that's okay.

02:34 That means this has to be zero as well.

02:36 So the f of zero equals zero by that way, by that definition.

02:40 So it works.

02:40 So it's defined at that point.

02:44 So that's good.

02:45 So now we need to show that the limit as, we'll use the definition of a derivative, the limit as h approach 0 of f of x plus h minus f of x all over h, that this limit needs to exist...

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