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7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

3 min readβ€’june 8, 2020


AP Calculus AB/BC ♾️

279Β resources
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πŸŽ₯Watch: AP Calculus AB/BC - Separable Differential Equations
Let’s look at the differential equation given in Eq. 12:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(960).png?alt=media&token=011b3f1e-f3e0-42dd-a3ec-2aa2c00eb52a
If we treat the derivative as a fraction, we can do something that would disgust professional mathematicians*:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(961).png?alt=media&token=5e424b6b-5e8e-4c01-8cb7-8a23391b999c
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:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(967).png?alt=media&token=1c2e412f-0fa2-4d0f-8197-655ec8e9b67e
Note that we need two constants because we found two different antiderivatives. However, we can combine them into a single constant:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(968).png?alt=media&token=66d73364-d781-4527-b7a6-11e7a49bb180
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,
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(969).png?alt=media&token=1450eba6-e9e1-46fe-b5c0-d0bf586e06ce
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:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(970).png?alt=media&token=e109d196-652a-426a-a850-34d1622bddd8
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:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(971).png?alt=media&token=fd801cf9-1682-4bed-8c23-f17687f8884e
We need to take special precautions regarding the absolute value signs when solving for y:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(972).png?alt=media&token=6c71e3e7-a2bb-4280-8dd9-9669c234ba0a
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:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(973).png?alt=media&token=f78536b2-4ebb-4434-9f38-72b2bf27776b
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.

Review

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.)
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(977).png?alt=media&token=b5ff0183-64fd-4415-bf60-11c6a12456d0

Answer

The arc trig expressions are red herrings:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(983).png?alt=media&token=be7a0a11-5a29-4812-b5ac-741cac992a27
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(981).png?alt=media&token=b8aa7977-2ff7-4001-a286-dbd7b7282c75

Footnotes

*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.
**β€œInfinite” 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:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(975).png?alt=media&token=8c4bd717-0f3a-4b34-a272-ca240d612405
Some definitions also include zero in the set of natural numbers (the lastly defined set in the statement).
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