5 min readβ’june 18, 2024

If you are good at **differentiating**, you will also be good at **testing possible solutions for differential equations**. Derivatives can be used to verrify that a function is a solution to a differential equation. 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. π€

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

To find *yβ*, 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**. βοΈ

Letβs try another example:

Here, weβll **differentiate** once, but weβll 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**.

Thus:

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 = -Β½ ****or ****x = Β½****. **

Here's a step-by-step guide to verifying solutions for differential equations:

- Write down the original differential equation: The first step in verifying a solution to a differential equation is to write down the original equation. This will give you something to compare the proposed solution to.
- Substitute the proposed solution into the differential equation: Once you have the original differential equation, you can substitute the proposed solution into the equation and see if it satisfies the equation. This is done by replacing all occurrences of the dependent variable (usually represented by y) with the proposed solution and then simplifying the equation.
- Check for an identity: After substituting the proposed solution into the differential equation, you should check to see if the resulting equation is an identity (i.e., the left side is equal to the right side). If the equation is an identity, then the proposed solution is a valid solution to the differential equation. If the equation is not an identity, then the proposed solution is not valid.
- Check the initial conditions: If the differential equation has initial conditions (i.e., specific values for the variables at a particular time), you should also check that the proposed solution satisfies the initial conditions. This involves substituting the initial values into the proposed solution and checking to see if the resulting equation is an identity.
- Test the solution: If the proposed solution satisfies the original differential equation and the initial conditions (if applicable), you can further verify the solution by testing it using different values of the variables. This can help to confirm that the solution is correct and not just a coincidence.

And believe it or not, this is not just a way for AP teachers to waste time in class. Verifying solutions to differential equations actually has many real-world applications. Verifying solutions to differential equations is an important process in many fields of science, engineering, and mathematics. Some examples of applications of this process include:

- Physics: In physics, differential equations are used to describe the behavior of physical systems. Verifying solutions to these equations can help to predict the future state of the system and understand how it will change over time.
- Engineering: Differential equations are also used in engineering to model and analyze complex systems. Verifying solutions to these equations can help engineers to design and optimize systems for a variety of applications.
- Biology: Differential equations are used in biology to model and understand the behavior of biological systems. For example, they can be used to study the spread of diseases or the population dynamics of animals. Verifying solutions to these equations can help biologists to make predictions and develop strategies to address problems in the field (if you've taken AP Bio, then you would recognize this process from the population model!)
- Economics: Differential equations are used in economics to model and analyze economic systems. Verifying solutions to these equations can help economists to understand how economic variables such as supply and demand interact and to make predictions about the economy. (Ap macro incorporates this!)

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*.)

- Not checking all the conditions: It is important to check that the proposed solution satisfies both the original differential equation and the initial conditions (if applicable). Failing to check one of these conditions could lead to an incorrect solution.
- Not simplifying the equation: After substituting the proposed solution into the differential equation, it is important to simplify the equation as much as possible before checking for an identity. This will help to ensure that you are not missing any terms or making any other mistakes.
- Not testing the solution: Even if the proposed solution appears to satisfy the original differential equation and the initial conditions, it is a good idea to test the solution using different values of the variables. This can help to confirm that the solution is correct and not just a coincidence.
- Not double-checking your work: It is always a good idea to double-check your work when solving any type of problem. Make sure to carefully review your calculations and solution to ensure that you have not made any mistakes.

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