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In Exercises $7-14,$ find the general solution of the first-order linear differential equation for $x>0 $$(y+1) \cos x d x-d y=0$$

$y=-1+c / e^{-\sin (x)}$

Calculus 2 / BC

Chapter 6

Differential Equations

Section 4

First-Order Linear Differential Equations

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A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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In Exercises $7-14,$ find …

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we have the differential equation. One place one times the call sign off X d X minus C. Why equal to zero so we can rewrite this US co sign off x times Why plus co sign off X minus the y the X equal to zero. We're these two terms here are just this product and we divide everything by the X And finally we get the differential equation. Why d X minus the call Sign off x times way, people to the course and off X. So here we will get P effects that is just minus a coursing X And this function here is going to be cure fixed. So now we can compute integrating factor that is given by your effects equal to e to the power off the integral off minus co sign if x d x Okay, seeing to growth p off x e x and if we saw that is going to be you tow the power of minus side affects DX. So that's integrated factor. And now that we have that we have that the solution is given by the function, why effects equal toe one over the integration factor? Oh, so Here's DC excess times the integral off cue effects time say integration, integrating factor plus the integration constant. Okay, so we can solve this integral here. Yes, yes. And this is our substation you equal to or let's just another letter w equal toe minus sign off X and what we'll get is he to sign off X That is this function here times minus E to the minus. Sign off X plus C We're this integral here. Give us us resolve that function. And if we make the product, we will get minus here to assign effects times he to the minus Sign off X plus c yeah, to sign off X We know that this product here is equal to one. So now we have that day solution off. The differential equation is a function c eight of the sign off pecs minus one

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