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For the following problems, determine how parameter $a$ affects the solution.Solve the generic equation $y^{\prime}=a x+y .$ How does varying $a$ change the behavior?

$y(x)=-a(x+1)+C e^{x}$Hence $a$ will have an effect if $C=0$ and its effect will control the slope of the line

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

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So for about five, number 2 57 we have an equation here with constant. So why prime is a X plus y? Let's solve this different equation and then see what the behavior of a could do for us of a changes. So to get this in standard form, why prime plus a function of X terms? Why is equal to another function of X So to get this guy in standard form that's going to be why prime minus why equals a X. The integrating factor here is going to be e to the minus one, the X So this is e to the minus X is the integrating factor. So multiply this equation by the integrating factor to get either the minor sex prime. It's equal to this a X e to the minus X. Then the key will be to integrate both sides of this equation On the left side, you have why e to the minus X and then on the right side. When you integrate this, you have minus okay and e to the minus x X plus one plus a constant of integration. Now, then, to solve this for why you have, why is equal to minus a X plus one plus see e to the X. So this is my different your equation. So what will happen? So obviously, if a zero, this is just an exponential function, but otherwise is the combination of a linear function and in exponential. So let's just take a look at what this looks like. So if I let a very Okay, so I'm a values air going between negative 10 and 10. You see how it changes the behavior of this graph? Okay, Now, you also see that, um, secret Beria's well, based on the boundary condition. So we've got several things going on here with his particular graph. Um, if you look at different values and then they could vary independently of each other different times, and so you get various behaviors based on how they are, how they're changing

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