First Order Differential Equations

I.  Homework

II.  Definition of First Order

Definition:  A First Order Linear Differential Equation is an equation that can be written in the form

y' + P(x)y = Q(x)

Example:  y' + y cot x = csc x

Here P(x) = cot(x) and Q(x) = csc(x)

We will show that y = xcsc(x)  is a solution of this equation.

Note that y' = csc(x) - xcsc(x)cot(x)

Substituting into the left hand side of the original equation:

[csc(x) - xcsc(x)cot(x) ] + [xcsc(x)]cot(x) = csc(x)

as required.

III.  The Solution to First Order Linear Differential Equations

Let y' + P(x)y = Q(x) be a first order linear differential equation.  The key to solving this equation is the magical integration factor

Notice that u' = P(x)u

Multiply the whole equation by u(x) gives

y'u + Puy = uQ or

y'u + u'y = uQ

Now notice that the left hand side is the result of the product rule on uy, hence

(uy)' = uQ

Now integrating both sides gives

uy = int(uQ)

or

IV.  Examples

Find the solution to xy'+ y = xe^x  

First we divide by x to get it into the proper form:

y' + 1/x y = e^x  

Now P(x) = 1/x and Q(x) = e^x  

the integrating factor is

u = e^{int Pdx} = e^{int 1/xdx} = e^lnx = x   

So the solution is

1/x int x² e^x dx

After integrating by parts twice we arrive at

y = 1/x (x²e^x - 2xe^x + 2e^x + C) = xe^x - 2e^x + 2/x e^x + C/x

We will do many examples from the book.