3 min readβ’june 8, 2020

π₯**Watch: AP Calculus AB/BC - ****Separable Differential Equations**

Letβs look at the differential equation given in Eq. 12:

If we treat the **derivative as a fraction**, we can do something that would disgust professional mathematicians*:

We can perform step 5 because the two sides of the equation are equivalent, thus we can perform the same operation (ie. integration) on both sides and not lose any information.

Continuing the **integration**:

Note that we need two constants because we found two different antiderivatives. However, we can **combine them into a single constant:**

This statement means that, because the set of real numbers is an infinite set** and both of the constants are real numbers, the difference between them is also a real number, ie. another constant we can call *C*.

So,

Notice that this equation is also equivalent to the solution given in Eq. 12. The solution given in Eq. 27 is a generalization of the solution given in Eq. 12.

Now, letβs say that we have some prior information about the function called an **initial condition**, eg. *y(0) = 1*. We can use this information to solve for *C*:

This implies that *C = 0*, which means we have found the specific solution to the differential equation in Eq. 12 along with the condition that *y(0) = 1*.

We can also revisit the differential equation in Eq. 11:

We need to take special precautions regarding the **absolute value signs** when solving for *y*:

So, the general solution is *y = C2e^x - 5*. For an additional exercise, find a specific solution to the differential equation such that *y(ln5)* *= -6*.

Another differential equation that is solvable via **separation of variables** is the equation below:

Notice that if *C = 0*, the solution to the differential equation is just *y = x*. If you plug *y = x* into the differential equation in Eq. 33, you will notice that this returns a valid equality.

Solve the differential equation below given an initial condition and a range restriction. (Hint: rewrite the arcsine expression using the identity that defines a cosine graph as a shifted sine graph and do something similar for the arccosine expression.)

The arc trig expressions are red herrings:

*This mathematical move is actually incorrect because multiplying the *dx* term over in that fashion implies that *dx* is a number and therefore *dy/dx* is a fraction, which is not true. However, this move is completely acceptable for the AP exam.

**βInfiniteβ in this context is not well-defined. The actual size of the set is called its cardinality, which is not simply infinity as implied in the statement:

Some definitions also include zero in the set of natural numbers (the lastly defined set in the statement).

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