1. Differential Equations Solving First-Order Linear DEs Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2. can always be put into the STANDARD form Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB The general solution will always have the form: y h is the solution to the corresponding homogeneous equation, where f(t)=0 y p is any particular solution to the original DE. There are five main ways to solve this type of equation - Immediate integration (has already been covered) - separation of variables (has already been covered) Other methods are: - Variation of Parameters (Euler-Lagrange Method) - Undetermined Coefficients (i.e. guess-and-check method only for Des with CONSTANT COEFFICIENTS) (- Integrating Factor Method, not covered in this course) First-Order Linear DE

3. Here are a few examples of linear, first order DEs. Put each of them in the standard format: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1) 2) 3) 4)

4. Here are a few examples of linear, first order DEs. Put each of them in the standard format: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1) 2) 3) 4)

5. Here are a few examples of linear, first order DEs. Put each of them in the standard format: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1) 2) 3) 4)

6. Here are a few examples of linear, first order DEs. Put each of them in the standard format: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1) 2) 3) 4)

7. Here are a few examples of linear, first order DEs. Put each of them in the standard format: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1) 2) 3) 4) Now that they are in the right format let’s try solving them.

8. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 2)Find the general solution to this DE.. This time let’s use the Euler-Lagrange method, also called “variation of parameters”.

9. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 2)Find the general solution to this DE. We use the Euler-Lagrange method, also called “variation of parameters”. This method is done in two stages. First we will solve the homogeneous version of the DE. Then we will use that solution to manufacture a particular solution to the original equation. Write down the associated homogeneous DE: This will be separable. Always.

10. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 2)Find the general solution to this DE. We use the Euler-Lagrange method, also called “variation of parameters”. This method is done in two stages. First we will solve the homogeneous version of the DE. Then we will use that solution to manufacture a particular solution to the original equation. Write down the associated homogeneous DE: This will be separable. Always. Here is the homogeneous solution. It will have an arbitrary constant.

11. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 2)Find the general solution to this DE. Now for stage two. We will use the homogeneous solution as a model for a particular solution. The trick is that instead of an arbitrary constant, we assume that there is an arbitrary function. We will substitute this into the original DE to make a new equation that we can solve for our new function v(t).

12. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 2)Find the general solution to this DE. Now for stage two. We will use the homogeneous solution as a model for a particular solution. The trick is that instead of an arbitrary constant, we assume that there is an arbitrary function. We will substitute this into the original DE to make a new equation that we can solve for our new function v(t). Quotient rule

13. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 2)Find the general solution to this DE. Now for stage two. We will use the homogeneous solution as a model for a particular solution. The trick is that instead of an arbitrary constant, we assume that there is an arbitrary function. We will substitute this into the original DE to make a new equation that we can solve for our new function v(t). Quotient rule We don’t need the constant of integration here. It would be redundant since we already have the homogeneous solution. There should always be some nice cancellation here.

14. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 2)Find the general solution to this DE. We are finally ready to put together our solution. This is what we found in stage 1.

15. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Euler-Lagrange Method - Variation of Parameters Here is a summary of the method: Write the Linear, First-Order DE in standard form: Solve the corresponding homogeneous equation: The particular solution will be the same as the homogeneous, but with the arbitrary CONSTANT replaced by an arbitrary FUNCTION of t. After taking the derivative and plugging into the original DE, you get a differential equation for v(t). The solution will be: Adding the homogeneous and particular solutions yields the answer:

16. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 3)Find the general solution to this DE. For this one we can start by finding the homogeneous solution, as if we are going to use variation of parameters. Then, instead of continuing with that method, we will make an educated guess to find the particular solution. First, the homogeneous equation:

17. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 3)Find the general solution to this DE. For this one we can start by finding the homogeneous solution, as if we are going to use variation of parameters. Then, instead of continuing with that method, we will make an educated guess to find the particular solution. First, the homogeneous equation: This should be a familiar solution by now.

18. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 3)Find the general solution to this DE. For this one we can start by finding the homogeneous solution, as if we are going to use variation of parameters. Then, instead of continuing with that method, we will make an educated guess to find the particular solution. First, the homogeneous equation: This should be a familiar solution by now. Look at the right-hand-side of the original DE and try to guess what the particular solution should be. This „GUESS AND TR”

19. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 3)Find the general solution to this DE. For this one we can start by finding the homogeneous solution, as if we are going to use variation of parameters. Then, instead of continuing with that method, we will make an educated guess to find the particular solution. First, the homogeneous equation: This should be a familiar solution by now. Look at the right-hand-side of the original DE and try to guess what the particular solution should be. We should try something like: A good first guess often uses something that looks like the right side, but with arbitrary constants.

20. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 3)Find the general solution to this DE. For this one we can start by finding the homogeneous solution, as if we are going to use variation of parameters. Then, instead of continuing with that method, we will make an educated guess to find the particular solution. First, the homogeneous equation: This should be a familiar solution by now. Look at the right-hand-side of the original DE and try to guess what the particular solution should be. We should try something like: A good first guess often uses something that looks like the right side, but with arbitrary constants. Plug this into the DE and you should get an equation for the arbitrary constant A:

21. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 3)Find the general solution to this DE. For this one we can start by finding the homogeneous solution, as if we are going to use variation of parameters. Then, instead of continuing with that method, we will make an educated guess to find the particular solution. First, the homogeneous equation: This should be a familiar solution by now. Look at the right-hand-side of the original DE and try to guess what the particular solution should be. We should try something like: A good first guess often uses something that looks like the right side, but with arbitrary constants. Plug this into the DE and you should get an equation for the arbitrary constant A:

22. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 3)Find the general solution to this DE. For this one we can start by finding the homogeneous solution, as if we are going to use variation of parameters. Then, instead of continuing with that method, we will make an educated guess to find the particular solution. First, the homogeneous equation: This should be a familiar solution by now. Look at the right-hand-side of the original DE and try to guess what the particular solution should be. We should try something like: A good first guess often uses something that looks like the right side, but with arbitrary constants. Plug this into the DE and you should get an equation for the arbitrary constant A: Now we have our answer: This method of “undetermined coefficients” will be used extensively for 2 nd -order DEs in Math 5A.

23. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 4)Find the general solution to this DE. Now that we have all the methods, try this one on your own. Use any method you like.

24. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 4)Find the general solution to this DE. Now that we have all the methods, try this one on your own. Use any method you like. The answer is: