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

a. 1b. 2

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

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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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