The first way to solve differential equations is by variation of parameters. This is an easy way to solve simple ordinary differential equations (ODEs) that involve only linear functions.
Variation of parameters works by finding the derivative of the equation’s parameter, then solving an auxiliary equation for the parameter instead of for the function. By substituting the parameter with its solution, one gets a new equation that can be solved.
Let us solve a simple linear ODE by variation of parameters! We will look at an example where y(t) = 1 + t − t2.
This ODE has no constant terms, so we can assume that any coefficient in front of the t−term is 1. Doing so leaves us with y(t) = 1 + t − t2, or y(t) = 1 + tv.
Solving a differential equation
Now, let’s solve for y as a function of x by variation of parameters.
We already found y when x is 1, so let’s keep it constant at 1. We will find the derivative of y at 1, and then find values of x where the derivative changes sign (where it becomes negative).
The derivative is Y” + 3y’ + 2y = 0 + 3(1) + 2(1) = -2
So when -2 = 0+3(1) + 2(1), then y=0 or y=2. The solution to the differential equation is {0,2}.
Variation of parameters
Another method to solve differential equations is by variation of parameters. This process involves finding an expression for the derivative of the function that is being solved for, changing parameter values to derive new parameters, and solving for the new parameters in order to find the solution.
More specifically, variation of parameters works by choosing a couple of constants and solving for their derivatives. Then, you choose new constants and variables that correspond to the original constants and variables, but are different enough to not be pure coincidence.
Then, you re-solve for these new variables using the original equation. If done correctly, you should get your original solution!
This problem is perfect for graph paper, so go out and buy some so you can do this properly.
Example: y” + 3y’ + 2y = 1 1 + Ex
The last example problem solved for y by solving a set of equations. This solution is not the most efficient way to solve for y, however.
Solving a set of equations requires one to find the value of y for each value of x, which can be time consuming. Also, if there are more x values then there are y values, then there is no solution!
Another way to solve for y is to use variation of parameters. This method uses one parameter to solve for another parameter instead of using two parameters to solve for one parameter.
Let’s look at an example: Suppose we have the equation y” + 3y’ + 2y = 1 1 + Ex where e = 2x Then we can use variation of parameters to find x:
Summary
In this blog post, you learned how to solve differential equations by variation of parameters. This includes how to recognize if a differential equation can be solved by variation of parameters, and how to solve the equation once you have the variable changes.
Solving differential equations by variation of parameters is a two-step process. First, you find the variable changes that solve the equation using some trial and error. Then, you solve the new equation that is created by changing the variables.
Variation of parameters is used when the function being solved for is not linear, meaning there is no line showing where zero is. When solving for a nonlinear function using this method, you have to check if your solution is constant or not.
References
This method to solve differential equations is by far the more common method. Variation of parameters methods are often taught in undergraduate courses as well as graduate courses.
They are also frequently used in industry as well as research. Because of this, it is important to be familiar with this method and how to implement it!
Solving differential equations by variation of parameters essentially breaks the differential equation into two separate ODEs: one for y’ and one for y. Then, we solve each of these separately, then put them back together using some operations.
The hardest part about this method is determining how to combine the two solutions. There are several tricks that help with this, which are mentioned above.
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