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Find the solution of the differential equation that satisfies the given initial condition.$ \frac {dy}{dx} = \frac {x \sin x}{y}, y(0) = -1 $

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$y=-\sqrt{-2 x \cos x+2 \sin x+1}$

Calculus 2 / BC

Chapter 9

Differential Equations

Section 3

Separable Equations

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

13:37

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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this question asked us to find the solution of the differential equation that satisfies the given initial condition we have D y over D backs is axe sign acts divide by. Why Now we know we we have to multiply both sides by why, in order to get all the y stuff or y terms on the left hand side, and then we know we can multiply both sides by D. X in order to get the ex terms on the right hand side like this. Not integrate. We know why becomes why squared over to us. We increase the exponent bar one divide by the new exponents and on the right hand side, we have negative acts Coastline X, plus sign, X pussy or constant. Now remember, we want this just in terms of singular Why? Which means we have to multiply both sides by two and then take the square root to get rid of the square root and the 1/2 as wth e coefficient we end up with Why is poster minus square root of negative two acts co sign acts plus to sign experts See, now we know they've given us an initial value of y of zeros. Negative. Want we can use this plug in wise negative one x zero we're solving for C. Don't forget. Once we saw first, see, we're gonna be plugging it back into the original equation. This simply means that seat is one. Because we know we're looking at simply negative one is negative square root of sea. We disregard the positive solution because that wouldn't logically sense. It only works if it's negative. Squirt of C two c is one now. Like I said, we're plugging the spot into the equation. We'd figured out why is negative square root of negative two acts co sign X plus to sign acts. The only difference is we're just adding our seed, which is plus one.

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