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In Exercises $1-10,$ solve the differential equation.$$\frac{d y}{d x}=y+3$$

$y=C e^{x}-3$

Calculus 2 / BC

Chapter 6

Differential Equations

Section 2

Differential Equations: Growth and Decay

Campbell University

Baylor University

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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Hello. We want to solve a differential equation given to us by D Y Divide by the X is equal y plus three. Yeah, if you'd like, you can have a look at example one in which, uh huh, a differential equations similar to this was solved by using the technique off separation of variables. We just mhm see. Okay. So the technique off separation of variables involves rewriting the equation if this is possible because it's not always possible. But if it is, this is the main. This is a good technique, a good method off solving differential equations. So we right, why functions of y and D Y on one side on the left side, let's say off the equation. And on the other side, we write expressions with X functions of X d of X on the other side. Then, after that, we integrate both sides with the appropriate with the appropriate mm integration variable. Here we have a few simple rules or a few simple basic integration formulas that you have at the beginning of the book with terror formula counts, for example, or earlier in chapters about integration. Chapter four, for example. Okay, so we go into this separation of variables business. Andi, what do we have here? We have the Y on the left side. We have one plus three on the right side. So we want to transfer this over there and this DX. We want to move it up here. How do we do that? We multiply the this equation with the X over y prostrate. So we're going to have d Y over y plus three is equal to the X on. Now we integrate Thistle is indefinite integration. So both sides should have after integrating after integrating. So I'll just write the intervals here. After integrating, we should have plus some constant here and plus some other constant here. But since constants can be transferred from one side of the equation to the other, we'll just write a constant on the right hand side. So it is going to be plus see here The right hand side is there is that This is what we use where a equals one on DT is DX, right? So what I have is the integral of one d X equals one times x plus c. So what I have is explosive like we have here on this side on this side. This is, uh we're going to solve this by using another integration formula. Basic integration formula that is going to be the integral off you over department. Do you? Over you is equal to L n off the absolute value of you. Plus food with a little substitution here with you Equalling. Why? Plus three, We have to find what d'you is going to be. Do you? Is there going to be equal to D Y zero? So the right hand side is going to be the integral off you Do you do you over you the integral off the u over you on. We know this to be equal to that. So l n off you, what is you? White plus three. Uh, plus constantly said we're only right constants on the other side. Explicitly. Now, if we want to explicitly, right? Why? As a function some function of f of X, we continue solving this to get rid of the natural logarithms on the left hand side of the equation, we are going to take ah, eight to the experiment off the left side to the exponents off the rights that were going thio exponentially. Eight The left on the right hand side. So and using the identity that a to the l n off some A is equal to a because the exponential function and the logarithmic function with based e r. Inverse functions of each other so e to the power off. Let me write it, then e to the power Eleanor Y plus three is equal e to the power explosive. Okay, now each of the power X plus e on the right hand side is can be written as mm e to the power off X times e to the power of C. All right, so this is some kind of some constant conflicts, and here we have Why? Well, three. Okay, So why is going to be equal to some constant? Let's call it if we go back. If we go back here, we can call this constant some other little D. But we want to use the sea, so we will go back and these constants off integrations that we added when we integrated here, we just call them instead of C will call them C one C one see one, C one and this. Some constant is going to be called C. So why is equal to see times e to the power of it, and this minus three will be transferred to the other side. So we are going to subtract three from both sides on this. Gives us, I think gives us, um, explicit functions, which is a growth or decay model, huh? But our job was just to solve the differential equation on. We will leave the interpretation for other problems that are here to come. So this is the procedure to follow. Separate the variables, apply the rules off integration or formulas for integrating. And then so the resulting equation for the variable. Why? If it's possible, sometimes it's not possible. Sometimes we will be forced. Sometimes we will be forced to, uh, right equation in an implicit way in which, why would would not be little like this, but it would be something an expression off. Expand why and see, Let's say something like this. There we have it. Hope it helps, but

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