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In Exercises $1-10,$ solve the differential equation.$$y^{\prime}=x(1+y)$$

$y=C e^{\frac{1}{2} x^{2}}-1$

Calculus 2 / BC

Chapter 6

Differential Equations

Section 2

Differential Equations: Growth and Decay

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

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In Exercises $1-10,$ solve…

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In Exercises $1-10,$ find …

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06:22

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01:42

Find $d y / d x$ in Exerci…

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Solve the differential equ…

Hello. Hello. Hello. Way need to solve this differential equation on the method. Basic myth that we are going to be able to use is the same method as in example one see example well, in which the method that we used was separation. The separation off variable. What is the separation of variables entail? We have an equation. And on one side of the equation, we will have variables. Why functions of why d y? And on the other we will have functions of it on the. And once we have separated the variables, we're going to integrate both sides by the appropriate variable. So the first thing we do is we're going to rewrite. This is d Y by the de y the so d y by the X is equal x multiplied one both. Why, in order to achieve the separation, we're going to multiply with the X and divide. With this print Mhm, we're going to multiply with DX and divide with one plus y. What we will have is d Y over one plus one equals experience. No, we integrate. Bye. Each of the by each of the integration variables. And I'm going to read them some of the rules that we're going to use on the left hand side. We're going to use the rules. Dear View U is equal to alien off the absolute value of you we'll see on the right hand side will be making use of extra para vein the integrated as X to the n plus one over that These Constance, we don't write on both sides of the equation just right. I'll just read them on the right hand side. So this is excellent. First, this is going to be X squared over two, and here we're going to have the U is equal to one plus y. So do you is the same d y. So here we will have the l n off. Why? Ln off one pass y is equal smoke further solving this for why we will apply the financial function e to the power off the left hand side and power of the right hand side since the end. Ln are inverse sanctions here we will have one plus y on here. We'll have e to the power of where the Let's see one. Now we apply it to the M plus in remember, we have some off funds. That means that we have a product off power. Right. This side is going to be written as e to the X squared of the two times E to the C one. This is one place for it. So since this is some sort off a constant, we are going to rename this constant to be seen. Um, we're also going to subtract one from either side, arrive to the final solution that y equals. See Time's E to the where money they were solution. The differential equation given food here in the beginning, I hope helped mhm, but

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