Cracking The Ap Calculus ab Exam 2018

There are many types of differential equations, but only a very small number of them appear on the AP
Exam. There are courses devoted to learning how to solve a wide variety of differential equations, but AP
Calculus provides only a very basic introduction to the topic.

SEPARATION OF VARIABLES

If you’re given an equation in which the derivative of a function is equal to some other function, you can
determine the original function by integrating both sides of the equation and then solving for the constant
term.

Example 1: If = and y(0) = 5, find an equation for y in terms of x.

The first step in solving these is to put all of the terms that contain y on the left side of the equals sign and
all of the terms that contain x on the right side. We then have y dy = 4x dx. The second step is to integrate
both sides.

∫y dy = ∫^4 x dx

And then you integrate

    =   2x^2    +   C

You’re not done yet. The final step is to solve for the constant by plugging in x = 0 and y = 5.

    =   2(0^2 ) +   C,  so  C   =   

The solution is = 2x^2 + .

That’s all there is to it. Separate the variables, integrate both sides, and solve for the constant. Often, the
equation will involve a logarithm. Let’s do an example.

Example 2: If = 3x^2 y and y(0) = 2, find an equation for y in terms of x.

First, put the y terms on the left and the x terms on the right.

    =   3x^2    dx

Next, integrate both sides.