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Solve the differential equations.Some of the equations can be solved by the method of undetermined coefficients, but others cannot.$$y^{\prime \prime}+y=\sec x \tan x, \quad-\frac{\pi}{2}<x<\frac{\pi}{2}$$

Calculus 3

Chapter 17

Second-Order Differential Equations

Section 2

Nonhomogeneous Linear Equations

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03:21

Solve the differential equ…

01:19

Find the solution of the d…

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Find the general solution …

section 16 that to problem number four were dealing with first order linear, ordinary differential equations. So in this case, we want to get everything in a standard form. So if I can write this differential equation is why prime plus some function of X times why is equal to some other function of X, Then I can multiply by an integrating factor, and that will allow me to solve this differential equation. Okay, this one is already in standard form, so my integrating factor is going to be to the integral of the tangent of X t X. And so this is going to be e to the minus natural log. Absolute value of CO sign of X now on the interval, negative part over to departure to the co sign is positive. So this just becomes he to the minus natural log co sign of X. So that's the co sign of X to the negative one power, which is, um, the Sequent. And so this turns out to be the seeking of X. So what would happen is I need to multiply this entire equation by the seeking of X. So the seeking of acts, Why prime plus a changing of X. Why is equal to co sign Squared X seeking of X. In doing so, the left side of this equation is just the product rule derivative of the seeking of acts. Times Why? So that would be the seeking of x d Y d x, and then plus seeking exchange in extent. Why so and then equal co sine squared of X seeking of X and we consider weaken Simplify that. So I've got derivative with respect to X of seeking X times y is equal to what I have. Cosan squared of x times the seeking of X, which is one over the coastline, which is just the co sign of X. So now when I integrate both sides of this equation, I get why times of seeking of acts is the integral of the co sign of X, the X, which is just sign of X plus c. So this gives us why is equal to the sign of X divided by the seeking of X plus C over the seeking of X and you wanna write this? You could rewrite this as sign of X one over the seeking its the coast on of X plus c co side of X, and that is the answer to our differential equation.

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