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# Solve the initial-value problem.$x^2y' + 2xy = \ln x, y(1) = 2$

## $$y=\frac{1}{x} \ln x-\frac{1}{x}+\frac{3}{x^{2}}$$

#### Topics

Differential Equations

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### Video Transcript

in order to solve this initial value problem. The first thing we can do is we can write this in terms of ax squared. Why now? Remember the product rule of differentiation F one G plus f g one. Everyone has a different way of writing this, but essentially in order to by using the product rule were able to understand how we can take the derivative and integrate both sides with respect toe acts. So we have the integral of one times natural log of axe D backs which gives us ax squared. Why is acts natural? Ivax minus the integral of D of axe. Now again, this is simply the integral of one, right? Which means we now have ex Corde. Why is act natural acts minus X plus c Lastly, dividing into these terms but explored in order to get why by itself we end up with this as our solution. Now we must plug in the values we have. Why one is two. I'm substituting in for X and why I have my ex. I have my why I console for my seat and I get my sea is to pull Swan which is three final solution written with the sea plug den because, like I said, we now know what R. C is would be thus.

#### Topics

Differential Equations

Lectures

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