Euler and Integration

[Keeley]

Right now we're working on Euler's and how accurate Euler's is at approximating a value of a function as opposed to integrating it.

One of the problems is this:

Given the initial value problem y'=0.5x(3-y) and y(0)=1.
Using integration, solve the differential equation to find the exact value at y(2).

The answer they give is . I literally haven't the slightest clue what to do.

So far, all I've done is distributed so that I get y'=1.5x-.5xy, and I don't even know if that's where to go with this.

 

[tkhunny]

Why would you do that? It's separable!

\(\displaystyle \frac{2}{3-y}dy\;=\;x\;dx\)

 

[Keeley]

Could you explain how you got to that because I still don't see it?

 

[tkhunny]

Change notation to the differential form: y' goes to dy/dx

Multiply by dx

Divide by (3-y) Note that y = 3 is bad and we'll have to keep an eye on this in any solution. (I also fixed the typo.)

Multiply by 2

The last was just for convenience.