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Problem 21 Hard Difficulty

Solve the differential equation and use a calculator to graph several members of the family of solutions. How does the solution curve change as $ C $ varies?
$ xy' + 2y = e^x $

Answer

$$y=\left[(x-1) e^{x}+C\right] / x^{2}$$

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Video Transcript

In order to solve this differential equation, we must divide by axe to get into standard form. Do you want I over d acts were dividing all terms by access. I said now that we have this, we can see our P is equivalent to two over acts, which means our each the integral of p of X eats the integral of two of her Axe de axe integrate. It eats the two natural of facts. This is a corpulent to eat the natural love with X squared. Remember E to the natural August 1. You get an interesting factor of simply x squared. Multiply each of the term of the differential equation by the interning factor. That's the next step. So I'm doing that right here. We can see that some terms end up canceling. So you have the same thing on top of the same thing on the bottom than you can cancel. Okay, Now we know we're gonna be using integration by parts, which is a concept in calculus that you should have learned by now. We have basically DVD you you envy So a cent. In other words, the formula for integration by parts is UV minus V Thames, the integral of you primes. In other words, you ve minus V times the integral of d'you. That's one way of writing it. So plug in or values you ve minus V times the integral off do you? So it's written like this, right? This simply in terms of why divide each of the terms by X squared, Because again, we want to get wide by itself in order to have an equation. Now it's saying to use a calculator, so we're gonna be using a calculator. We could simply plug this end, and they want us to look at the members of the family. As you can see over here, when C is less than zero seemed being the integrating factor, the curve decreases, and then we no one sees greater than zero. The curve increases when C equals to one. It looks like it crosses The y. Axis, however, is actually undefined. X equals here, which means there was a hole, which is another accomplice concept that you've probably covered. This is the equation that we covered earlier. Why is X minus one times E to the X pussy over X squared