MATH 046 Lecture 9: Beginning Bernoulli Equations

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graygnu278

14 Dec 2016

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+ p(x)y = q(x)y y n for some number n. this is not a linear equation, unless n is 1 or 0. However, by looking at a new unknown function related to y, we can transform it into a linear equation. + 2y = e y x y. We consider a new unknown function z = 1 y. We shall derive a di erential equation for z based on the di erential equation for y as follows: z. E x = 2z e x. Find the linear equation for z = 1 y. As a result, z satis es the linear equation z.

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2d and 2f

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First, create 3 equations of the form ax+by+cz=d , where a, b, c, and d are constants (integers between – 5 and 5). For example, x + 2y – 2= -1 . Perform row operations on your system to obtain a row-echelon form and the solution. Go to the 3D calculator website GeoGebra and enter each of the equations. After you have completed this first task, choose one of the following to complete your discussion post. 1. Reflect on what the graphs are suggesting for one equation, two equations, and three equations, and describe your observations. Think about the equation as a function f of x and y, for example, x + 2y +1 = 2 in the example above. Geogebra automatically interprets this way, that is, like z=f(x, y)=x+2y+1 , it isolates z in the equation. 2. What did the graphs show when you entered the second equation? 3. Give a simple description of the system x=0 y=0 z=0 x = 0 can be seen as the constant function x=g(y, z)=0y+0z=0 . Of course, you can use GeoGebra to “observe” the system. 4. Give an example with 2 equations as simple as possible with 3 variables (at least 1 being non-linear; keeping z to the one power on both equations) and describe the potential of GeoGebra to study nonlinear systems.

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MATH 046 Lecture 9: Beginning Bernoulli Equations

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