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. 🤔
For example, we will test the following solution for the succeeding differential equation:
To find y’, we shall rewrite y in the form of a power:
Following this step, we take the derivative by the power rule, followed by the chain rule:
Taking the derivative induced by the chain rule and simplifying the expression, we find:
Thus, the differential equation holds. ☑️
Let’s try another example:
Here, we’ll differentiate once, but we’ll use a technique that significantly cleans up the problem instead of jumping straight to differentiating again:
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:
The substitution made in step 3 in Eq. 18 is a combination of the product rule and implicit differentiation.
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:
The substitution made in step 2 is made possible by Eq. 16. The above condition holds only for x = -½ or x = ½. ❗️
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.)