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7.2 Verifying Solutions for Differential Equations

#solutions

#differentialequations

โฑ๏ธย ย 2 min read

written by

Jacob Jeffries

jacob jeffries


If you are good at differentiating, you will also be good at testing possible solutions for differential equations. The process is as simple as taking a derivative, plugging it into a formula, and then doing some simplifying to show that the solution does or does not work. ๐Ÿค”

For example, we will test the following solution for the succeeding differential equation:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(913).png?alt=media&token=3f50ea56-ba8a-47cf-8496-8ab9dc340ca2

To find yโ€™, we shall rewrite y in the form of a power:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(914).png?alt=media&token=1a5fd919-6bec-45cf-9e09-e9216a88deff

Following this step, we take the derivative by the power rule, followed by the chain rule:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(915).png?alt=media&token=833d707f-cdfe-4869-a2db-61eebb68bf45

Taking the derivative induced by the chain rule and simplifying the expression, we find:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(916).png?alt=media&token=ab4c61e3-501a-470b-bcca-32c05187d600

Thus, the differential equation holds. โ˜‘๏ธ


Letโ€™s try another example:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(917).png?alt=media&token=87c6f572-7b73-4a6b-ad0f-b294612aaa21

Here, weโ€™ll differentiate once, but weโ€™ll use a technique that significantly cleans up the problem instead of jumping straight to differentiating again:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(918).png?alt=media&token=9a36dd41-1d1d-4972-867f-3c6f6b2c78c1

The substitution made in step 2 in Eq. 17 (as denoted by the 2 over the equality sign) is made possible by Eq. 16.

We can now use the product rule to find the second derivative:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(919).png?alt=media&token=2909db25-0e9a-4572-8290-2a4fee719cd5

The substitution made in step 3 in Eq. 18 is a combination of the product rule and implicit differentiation.

Thus:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(923).png?alt=media&token=2b32cb86-3133-423f-805a-13494f16cdbe

The substitution made in step 1 is made possible by Eq. 17. This method may seem convoluted and unnecessary, but it seriously simplified some very tedious algebra.

From here, we can already see that the differential equation will not hold for all x. However, we can see which value(s) of x it will hold for by plugging it into the differential equation put forth in Eq. 16:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(920).png?alt=media&token=d8470324-4725-4c21-9180-9dc086390fd1

The substitution made in step 2 is made possible by Eq. 16. The above condition holds only for x = -ยฝ or x = ยฝ. โ—๏ธ

Review

Test if the function given in Eq. 20 is a solution to the differential equation in Eq. 21. Find an expression for the max value of N and the maximum growth of N. (Hint: use graphing software to find a shortcut to finding the max value of N.)

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(921).png?alt=media&token=a8f6e5a4-3182-4c52-842b-5c0f78d42ca9

Answers

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(927).png?alt=media&token=fc168f99-650f-4602-9e8d-0b46a685b028

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(925).png?alt=media&token=1910337a-38a0-4e91-b61b-5d73219fbbfc

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