Differential Equation Review In this course you will be expected to be able to solve two types of first order differential equations without Maple just as you are expected to be able to do basic calculus without Maple The first type of differential equation to be reviewed is a separable differential equation and the second is a differential equation that can be solved using an integrating factor Once you understand how to deal with these simple types of equations and how they affect system properties generalizing to other types of differential equations will not be difficult Separable Equations A separable differential equation is generally one where we can rewrite the equation so all of the variables on one side of the equation are the same In what follows we will assume the system starts at time to and ends at time t Example 1 Consider the separable differential equation x t t 2 We first rewrite dx dx the derivative as x t so we have t 2 Next we put all of the x s on one dt dt side of the equation and all of the t s on the other side of the equation so we have dx t 2 dt Now we want to integrate both sides of the equation At time to x has value x to while at time t x has value x t Hence we have x t x to t dx 2 d to Note that we are using a dummy variable of integration so there is no confusion with the end points of the definite integral Specifically if we were to use t as the dummy variable of integration and as one of the endpoints of the integral we would most likely make mistakes Performing the integration we get x t x to t 3 to3 3 3 Finally we get the solution x t x to t 3 to3 3 3 Example 2 Consider the separable differential equation y t 2t y t We first dy dy rewrite the derivative as y t so we have 2ty t Next we put all of dt dt Fall 2006 the y s on one side of the equation and all of the t s on the other side of the dy equation so we have 2tdt Now we want to integrate both sides of the y equation At time to y has value y to while at time t y has value y t Hence we have y t t dy y t 2 d y to o or y t 2 2 2 2 ln y t ln y to ln t to t to y to Finally we get the solution y t y to e t 2 to2 Example 3 Consider the separable differential equation y t 3 y t We first dy dy rewrite the derivative as y t so we have 3 y t Next we put all of dt dt the y s on one side of the equation and all of the t s on the other side of the dy 3dt Now we want to integrate both sides of the equation so we have y equation At time to y has value y to while at time t y has value y t Hence we have y t t dy y t 3d y to o or 2 y t 2 y to 3 t to Finally we get the solution 3 y t y to t to 2 Fall 2006 2 Differential Equations with Integrating Factors An integrating factor allows us to write one half of a first order differential equation as an exact derivative something easy to integrate and the other part as a function with no derivatives In what follows we will assume the system starts at time to and ends at time t In the first example we go over all of the details but in the final two examples we just use results from Example 4 Example 4 Consider the differential equation y t y t 2 We first rewrite the dy dy derivative as y t so we have y t 2 Next we want to write the left dt dt d hand side of the equation as y t e a t Using basic properties from calculus dt we have d dy t da t a t dy t da t y t e a t e a t e y t e a t y t dt dt dt dt dt We want the term in the brackets to look like our original equation that is we want dy t da t dy t y t dt dt y t dt Equating the two sides we get da t 1 dt which gives us a t t At this point we have d dy t y t y t e t e t dt dt Now since we have from our original differential equation dy y t 2 dt we can multiple both sides of this equation by e t to get dy t e t y t e t 2 dt or Fall 2006 dy t d y t y t e t e t 2 e t dt dt At this point we have the left hand side as an exact derivative d y t e t 2e t dt Now we want to integrate both sides of the equation t t d t t dt y t e dt t 2e d o o Integrating we have y t e t y to e to 2e t 2e t0 Finally we get the solution y t y to e t t o 2 2e t to Note We can also solve this equation in the same way we solved the separable equation by going through the following steps dy dt 2 y y t t dy d 2 y t o y to 2 y t ln 2 y t ln 2 y to ln t to 2 y t o 2 y t e t to 2 y to 2 y t 2 y to e t to y to e t to 2e t to y t y to e t to 2 2e t to Example 5 Consider the differential equation y t 2ty t x t From Example 2 2 da t d 4 we need 2t or a t t 2 We then have y t e t e t x t dt dt Integrating both sides we have Fall 2006 t t d t 2 2 t dt y t e dt t x e d o o or t y t e t y to e to x e d 2 2 2 to Finally we have the solution y t y to e t 2 to2 t x e t 2 2 d to Note that we cannot go any further in the solution until we know x t 3 t y t et x t From 2 3 3 3 d da t 3 t2 t2 t 2 Example 4 we need t or a t t We then have y t e …
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