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In Exercises $1-10,$ solve the differential equation.$$y^{\prime}=\frac{5 x}{y}$$

$y^{2}=5 x^{2}+C$

Calculus 2 / BC

Chapter 6

Differential Equations

Section 2

Differential Equations: Growth and Decay

Oregon State University

Baylor University

University of Nottingham

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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Solving a Differential Equ…

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In Exercises $1-10,$ solve…

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Hello. I'll jump here is to solve this differential equation. Uh, look, we owe See, the method we're going to be using is the same as the method. In example one which is the method off separation off variable. What this method involves, if possible, is to have any equality sign And then on this side of the equation, we have some, uh, wise functions of wide and the wise, and here we have x functions of it. The leaks possible. So what is the derivative off? Why? Yeah, d y by the So that's the first thing that we're going through, right? D Y by D. X is equal to for a ah, by the way, we would like to associate why with the y and D x five x So what we're going to do is multiply this with why the so we will have y d y is equal to five mhm variables separated. What we do now is we integrate Well, if and the right so let me integration side. This is going to be indefinite integral, so they wouldn't involve a constant as a result of integrating on both sides. But Constance can be shifted to the other side. So when we when we integrate, we're going to write constant Onley on one side and we'll call, see one for the time being. Have to see Maybe father Constancy emerged so we can have a final concept of C right now. A Sfar Azaz integrating his concerns. We're going to be using basic integration formulas that you can find in the cutout section of your book or in the appendix, or in the chapters that dealing with integrations that preceded this one food Oh, both integral are one and the same. So if we have if we have a eight times f o x e x, if we have this integral them, this is equal to eight. This is the cost of multiple. Okay, what? This five does little jump in front of integral on the second interval is if we have a variable t times the tea, then it integral the weird over to plus see or if we have any. If we have any power off teeth, we had t to the power in the t. The interval will be t where n is increased by one over and passed one, so integrating the left side we have. This is why squared over two. Okay. And here we have five times it x squared over to flu eso multiplying the equation with two we have one squared is equal to five x where plus to see one this we really find as a constant see on we will Instead of giving an option of taking the square with plus minus square root off the function We are going to implicitly give the track the five X script from both sides y squared minus five Weird is equal. Okay. Functions that satisfy this vision our solutions Thio the differential equation coming up here. There we have it. Hope it helps, but

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