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Solve the differential equations subject to the given initial conditions.$$\frac{d^{2} y}{d x^{2}}+y=\sec ^{2} x, \quad-\frac{\pi}{2}<x<\frac{\pi}{2} ; \quad y(0)=y^{\prime}(0)=1$$

Calculus 3

Chapter 17

Second-Order Differential Equations

Section 2

Nonhomogeneous Linear Equations

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02:05

Solve the following differ…

05:57

Solve the differential equ…

13:35

04:45

Find the particular soluti…

00:33

Find a solution of the sec…

So start off this problem by rewriting Y. Prime as Dy developed by T. T. That equals to Y squared sign time. Scientific. And so now let's divide it by y squared on both sides. So I have that the Y. Sorry why squared DY equals two negative sign T. Teaching. You can integrate both sides here and this will be negative one over why On the right hand side was gonna integrate to co sign T. Plus C. So we can take the reciprocal of both sides will have that negative Y equals two. One divided by ko 70 pussy. If you multiply both sides by negative one we'll have that one over C minus co sign T. And now we can use the initial conditions that we have which is why a pi over two equals one. So where we see a white and we're gonna plug in one and where we see a teacher and we're gonna plug in pi over two. So we're going to have co signed harbor too. And this entire term goes to zero Mexico sign of pirate two is zero. And so we have that one equals to one oversee rather that C equals to one. And so with this, we can actually write our final solution. Her final solution is why equals two one divided by one minus co sign T. And we can rewrite this if we want to be Y equals two negative one divided by co sign t minus one. So that will be your final answer.

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