Explain why or why not Determine whether the following stateme...

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The differential equation y′+2y=t is first-order, linear, and separable.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says consider the differential equation W plus 5 W is equal to U cubed. Which statement is true? And we're given 4 possible choices as our answers. For choice A, we have the equation is first order, linear, and separable. For choice B, we have the equation is first order and linear, but not separable. And for choice C, we have that the equation is not linear. And finally, for choice D, we have the equation is separable but not linear. So the first thing we want to do is determine the order of our differential equation. And if you look at our differential equation, the highest order of derivative that we have is the first derivative, since we just have W in our equation. That means that this is going to be a first order. Differential equation. Next, we need to determine whether it's linear or not. And here we need to look at the highest power of W. So, if we look at our equation, the highest power of W that we have is W to the first power, which means that this here is a linear equation. And finally, we need to determine whether this is separable or not. And recall that in order for the equation to be separable, we need that W is going to be equal to some function of W, which we'll call F of W. And it needs to be multiplied by some function of U, which we'll call G of U. So, if we take our equation here, And solve it for, we'll see that we have that W is equal to. You cubed minus 5 W. Which, if we wanted to write this in terms of functions of U and functions of W, this would be equal to. G of you. Minus F of W. Which is not equal to. F of W. Multiplied by G of U. And so that means that this equation is not separable. So what we found was that this differential equation was a first order linear differential equation, but it is not separable. And so, that's the answer that we're looking for for this problem. If we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice B. So, I hope this explanation has been useful, and I'll see you in the next video.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. The differential equation y′+2y=t is first-order, linear, and separable.

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The differential equation y′+2y=t is first-order, linear, and separable.