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Solve the following initial-value problems by using integrating factors.$$y^{\prime}+y=x, y(0)=3$$

$y=x-1+4 e^{-x}$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

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Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

02:36

Solve the following initia…

04:20

03:44

03:48

03:23

02:42

Solve the following differ…

03:03

05:25

08:03

Determine an integrating f…

01:54

Solve the initial value pr…

so a section 4.5 problem to 41. We're dealing with a linear, ordinary first order differential equation, and we have a boundary condition. 50 is equal to three on. So what we need to do is just to put this in standard form. And we say, Well, it's already in standard forms. A standard form is why prime some function of X times. Why is equal to some other function of X so I could find the integrating factor is going to be e and so the coefficient of y just one. So this is going to be e to the X is the integrating factor. So that tells me I can rewrite this differential equation as why into the X prime is equal to X e to the X, Then the solution would be to integrate both sides of this equation. So in integrating both sides of this equation, I end up with why e to the X is equal to either the ex text minus one plus a constant of integration then to solve for why it's just a divide by either the x o why is equal to X minus one plus c e to the minus X. Then you have the boundary condition that why of zero 2nd 3 that tells me that to solve for C, I'm gonna have a three. It's a people to zero minus one. See? And then each of the zeros just simply one. So that tells me that C is equal to four. So the final answer here is why is equal to X minus? One was for E to the minus X, and that is the exact solution to this differential question.

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