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Find the solution of the differential equation that satisfies the given initial condition.

$ \frac {dy}{dx} = xe^y, y(0) = 0 $

$$y=-\ln \left(1-\frac{1}{2} x^{2}\right)$$

Differential Equations

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this question asked us to find the solution of the differential equation. Given D Y over, G axe is exceeds the why. First things first, we know we wanna put the Y stuff from often side either negative want why do y than the X stuff on the right hand side? Because now we can much more easily integrate. Integrating loft inside was simply out of negative sign. In trading the right hand side, we increase the exploited by one and then we divide by the new exponents. Now we know we're gonna be substituting an X and Y are both zero and solving for C because they've asked us that in the problem we got C is negative one. Substitute the value of C back end of negative one and simplify. We know we end up with negative. Why is natural log? We get not your logs. When you do natural of times eat, you cancel off and you simply end up with one. So that's why the E goes away is because natural times he's won. Then lastly, we want this in terms of positive. Why not negative wife? So multiply by the negative sign