Differential Equations

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

A First-Order Linear Differential Equation can always be put into the form: The general solution will always have the form: yh is the solution to the corresponding homogeneous equation, where f(t)=0 yp is a particular solution to the original DE. We will have four main ways to solve this type of equation. One of them, separation of variables, has already been covered. The other methods are: *Integrating Factor Method *Variation of Parameters (Euler-Lagrange Method) *Undetermined Coefficients (i.e. guess-and-check method) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

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

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

1) Find the general solution to this DE. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. We have several options for this one. The equation is not separable, but we can use an integrating factor, or variation of parameters. Let’s try the integrating factor. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. We have several options for this one. The equation is not separable, but we can use an integrating factor, or variation of parameters. Let’s try the integrating factor. First we have to calculate our factor [call it µ(t)]. Here is the formula: So we have to integrate the function in front of the y in the equation. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. We have several options for this one. The equation is not separable, but we can use an integrating factor, or variation of parameters. Let’s try the integrating factor. First we have to calculate our factor [call it µ(t)]. Here is the formula: So we have to integrate the function in front of the y in the equation. Don’t worry about the constant of integration here, it won’t matter. Once we have this integrating factor, the trick is to multiply it through the original equation, and then stare at the left-hand side. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. Here is the equation after we multiply through by the integrating factor: Look at the left-hand side and try to figure out the trick. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. Here is the equation after we multiply through by the integrating factor: The left-hand side will always be the derivative of a product. Apply the product rule to the left side and prove it to yourself: At this point we simply have to realize that if we integrate both sides, we can then divide by the integrating factor and we have it solved. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. We will need to integrate the right side (the left side is already done) Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. We will need to integrate the right side (the left side is already done) Integrate by substitution: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. We will need to integrate the right side (the left side is already done) Integrate by substitution: Now divide to solve for y: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

1) Find the general solution to this DE. We will need to integrate the right side (the left side is already done) Integrate by substitution: Now divide to solve for y: This is the general solution. Notice that it has 2 parts added together. These are the homogeneous and particular solutions that we expected. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Integrating Factor Method Here is a summary of the method: Put the Linear, First-Order DE in standard form. Next, calculate the Integrating Factor. You can multiply through the DE. It should look like this: Now you recognize that the left side IS the derivative of µy. Integration yields: This constant of integration is super-important. It is where the homogeneous part of the solution comes from. You can go straight to this final formula if you like. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

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

2) Find the general solution to this DE. Again we have several options for this one. This time let’s 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. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

2) Find the general solution to this DE. Again we have several options for this one. This time let’s 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. 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). 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 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 There should always be some nice cancellation here. We don’t need the constant of integration here. It would be redundant since we already have the homogeneous solution. 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. 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: 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: 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. 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. 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. 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: 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 2nd-order DEs in Math 5A. 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. 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: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Differential Equations