slope fields
example:
For a graph where
dydx=0.4y
The first step is always to make a graph of the slopes at multiple integers of x and y.
so if we use the specific point (2,4), we can integrate
and find a particular solution to the differential equation y=2e^(.4x).
euler's method
1. FInd slope by dy/dx
2. Add dx to x.
3. Multiply slope by dy/dx by dx.
You now have dy
4. Add dy to y
5. Repeat until the appropriate x-coordinate is reached and you will have a full coordinate set, both y and x.
2. Add dx to x.
3. Multiply slope by dy/dx by dx.
You now have dy
4. Add dy to y
5. Repeat until the appropriate x-coordinate is reached and you will have a full coordinate set, both y and x.
Take Euler's method as an example. Given that
y=e^x dx=.1 and the initial point is (2,1), find y when x=10
y=e^x dx=.1 and the initial point is (2,1), find y when x=10
Thanks to mathscoop.com, we can use their free Euler's Method calculator to graph the equation and see the iterations of adding dx to each x and the dy to each y.
Keep in mind that this is a very lengthy example and we can only hope that the AP® Calculus test will never be this cruel.
Keep in mind that this is a very lengthy example and we can only hope that the AP® Calculus test will never be this cruel.
separation of variables
Given a relationship of functions, you must solve the equation by tearing the traditional "slope" limb from limb by multiplying by dx.
For example
dy/dx= 3x^2 * cosy
dy= 3x^2 * cosy dx
divide by the y-term...
cosy dy = 3x^2 dx
and integrate with respect to nothing because you already have dy and dx.
S cosy dy = S 3x^2 dx
siny = x^3
y= sin^(-1)(x^3)
You have found the equation y, congrats! Now you can do cool things like plug it into your calculator or graph the equation.
For example
dy/dx= 3x^2 * cosy
dy= 3x^2 * cosy dx
divide by the y-term...
cosy dy = 3x^2 dx
and integrate with respect to nothing because you already have dy and dx.
S cosy dy = S 3x^2 dx
siny = x^3
y= sin^(-1)(x^3)
You have found the equation y, congrats! Now you can do cool things like plug it into your calculator or graph the equation.