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Verify that $y(x)=e^{x} \sin x$ is a solution to the differential equation$$2 y \cot x-\frac{d^{2} y}{d x^{2}}=0$$

$y(x)=e^{x} \sin (x)$ is a solution

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 1

Differential Equations Everywhere

Differential Equations

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and this problem were provided with the second order differential equation on her job is to show that why I've X equals either the X sign X is a solution. Our first job is to take the derivative of this given function well, right that D y dx will be equal to by the product rule first, a derivative of even the X copy sine X plus copy either the ex itself and now take the derivative of sine X, which turns into co sign X. Now, when we take the derivative of this function, we obtain detail Why divide by DX to and we go term by term. When we take the derivative of this first term, However, we note that was the original work we did on the first step in finding the first derivative. So it's just copy down. This result, the driven of even the X sign X is either the ex sine x plus. You know the X Times Co sign X. Now let's go to the very next term. Either the ex co sign X when we take this derivative will add either the ex copy co sign X. Let me add a little bit of space, a copy of the X and take the derivative of co sign Cosan becomes negative sine X salts Insert the negative between these functions and right to minus either the ex sine X And that's our second derivative. Let's simplify the second derivative as much as we can their very first term either The ex I necks cancels with the last term. And so the result is two copies of either the ex Times Co sign X. Now we have a second or derivative. We've completed the second derivative. So the next phase of the problem is to determine if this left hand side does, in fact equals zero through substitution. Denote this LHs to stand for left hand, side and right that the left hand side is equal to two times the function. Why? But why is given to be, you know, the X sign X sold do two times either the ex sine x Times co tanja necks minus due to why DX to that was the last step that we calculated on the second derivative which came out to to either the ex co sign X. So we get to you know, the X co sign X, and we've completed all of our substitution so far. Recall our goal is to now simplify as much as possible obtaining zero. If we can get to that step on the right hand side, then we'll know that we have a solution. So start off by baby first expressing co tangent into this components co sign of X and sign of X. We have two times you the X sine x co tension itself turns into co sign X divide by sine X, and we have minus two either the ex co sign X by copying another cancellation with the sign Exes leaves us with two of the X coastline X minus two to the Exco sine x So we do it in fact get zero. And this then shows that why of X equals the X sign X is a solution.

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