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State the order of the differential equation, and confirm that the functions in the given family are solutions.$$\begin{array}{l}{\text { (a) } 2 \frac{d y}{d x}+y=x-1 ; \quad y=c e^{-x / 2}+x-3} \\ {\text { (b) } y^{\prime \prime}-y=0 ; \quad y=c_{1} e^{t}+c_{2} e^{-t}}\end{array}$$

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Calculus 1 / AB

Calculus 2 / BC

Chapter 8

MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS

Section 1

Modeling with Differential Equations

Integrals

Integration

Integration Techniques

Differential Equations

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

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.

02:04

State the order of the dif…

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02:28

09:12

Verify that the indicated …

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10:26

01:54

$y=c_{1} e^{x}+c_{2} e^{-x…

02:53

Okay, so here we are to state the order of the given different recreation here, we have to do. Right Next. Plus y is equal to X minus one. Let's call this. Well, let's call this one. And within our to confirm that the functions in the family why is equal to see times e to the negative x or two plus X minus three. Um, are solutions that's called this'll here, too. Now, looking at one right, since the first derivative is the highest derivative that occurs, right, we only have a divide the after two times do RDX, where we have a first derivative here. There are no higher order derivatives to no second to revenues the derivatives or so on. So the highest derivative we have is a first derivative. Therefore, the order of his defensive question is one. So the order is one. Okay, Now we go ahead and we differentiate to here so he gets de y The axe is equal to see. This is gonna be negative. 1/2 see me to the negative X over two, uh, plus one. Okay. And, um, so, um, where we have to the wide yaks. Plus why is equal to chew times negative 1/2 C e to the negative X over two plus one plus I c e to the negative X over two plus X minus three. Okay, so we have, um, is equal to C E to the negative X over two plus two c e to the negative X over two plus X minus three, which is equal to X minus one. So, therefore confirm that, too is a family of solutions for one and then forward party. Well, we have difference equation. Here we have. Why double crime minus why is equal to zero case. Let's call this one. And we are to confirm that the functions in the family why is equal to see one? Yeah, um, to the, um each CCI so see why I e too t plus C two eats too. Negative to you, um, are our solutions. So let's call this here too. Now, looking at one again, we have a second derivative here. Why? Double crime minus Why so since the second derivative is the highest derivative in this different equation, right, the order is to So the order of a deflection equation is just the highest. Um, highest order derivative that occurs. Just we have a second derivative here. The order of this difference equation is too. Okay. Now, looking at, um, were you created here? We go ahead and we differentiate both sides. So we get d Y d t 28 with respect to T. Um, well is equal Chew. Um, well, that's why prime right is equal to see one the t minus C two e to the negative t and we differentiate again and we get why double crime is equal to see one e to the t plus C to need to the negative. Yeah, um, which is equal to why? Which implies that why the prime minus y is equal to zero. So therefore, it is confirmed that two right, that, um to here is a family of solutions of one. Okay.

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