Let’s 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.
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, let’s 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.
**“Infinite” 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).