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Solve the following initial-value problems by using integrating factors.$$y^{\prime}=2 y+x e^{x}, y(0)=-1$$

$y(x)=-(x+1) e^{x}$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

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Lectures

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Solve the initial value pr…

04:12

Solve the differential equ…

Israel. We are going to solve this differential equation. Why? Prime minus two y equal x times e to the X with initial condition that X equals zero give Why equal minus one. Okay, um so it is stand out. Form P of X is just minus two. Integrating factor would be e to the minus two x We multiply this true left hand side going to become Why times e to the minus two x Oh, this primes equals right hand side will be here will be e to the minus X now And we have to integrate the right hands I So we use it to grow by past again. You is just x Do you is e to minus x d X Now the U is the X and we will be minus E to the minus eggs. Put everything in the formula we get, um to the end. It will be this. So this should be where if you should be very familiar with this technique already, and the term that you see Owen over like x time e to there's something should be to it too hard. So practice. Ah, we put this in back in here, and I'm gonna skip to what? Why? Looks like here. So we shared from E to the minus to the, uh, plus X now, because we we used to have even a mite has two x on the left side. And like we calculate it out, we get this general solution. Now, we put in the condition here. Why is minus one when x is zero? So is zero minus one plus c. So the sea is actually Ciro this term gonna disappear and yeah, why will be just this, um, I thing You can pull this out. Yeah. Um sorry. Let me shake. Yes is correct. Uh, this is Thea Answer. There's no constant yet because C is zero. That is the answer. Thank you.

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