7.2 Verifying Solutions for Differential Equations

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1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

1. 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.
2. 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.
3. 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!)
4. 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!)

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Answers

1. 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.
2. 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.
3. 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.
4. 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.