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Solve the given differential equation.$\frac{d y}{d x}=a x^{N} y^{M}, a$ is a constant. (Note: there are four cases that need to be considered.)

$$\begin{array}{l}M=1, N \neq-1 y=c e^{\frac{a x^{N+1}}{N+1}} ; M=1, N=-1 y=c\left|x^{a}\right| \\M \neq 1, N=-1 y^{1-M}=a(1-M) \ln |x|+c \\M \neq 1, N \neq 1, y^{1-M}=\frac{1-M}{1+N} a x^{N+1}+c\end{array}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Campbell University

Idaho State University

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Solve the given differenti…

00:51

Solve the differential equ…

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Hello. Let's get to solving this problem, shall we? So, step one, we just find integrating. Factor in the reading factor is equal to e to the integral of em over X, which in this case, is when we eat it integral of p of X and p of X is one in front of why? So you know, that's where we ended with MX. Over here. Memorex so integral of each of the Memorex, the same thing as e to the integral. We pull the amount of one of her ex. This turns into E to the, um l and of X deliveries lager Evanses, ankles of top centipede to the Ellen of extra G m, which just becomes heated. L or something, which is X to the m. There we go. Easy now I want to do is multiply both sides by interviewing factory. So we end up with exit e m. Why Prime plus M over X. Why is equal to X m l f of X? Easy now he's needs to the interval sides. So for the 1st 1 we just end up with white hunter integrated factor, which is just gonna be why Times X City am There we go. Because the divert to of white times xdm Well, give us this. There we go. Meal all the right hand side We end up taken integral of x city AM Ellen of X Only to do is a integration by parts. This part will be Devi. This part will be you integrate with the formula You times vm I's integral sign of v times, do you? So as we go on, we end up with you which is Ellen of X V, which is the integral x and M, which is exit E m plus one all over em plus one minus. Integral Sign V, which is going to equal exited M plus one over M A plus one times the U The Ritter view is one over X. There we go. These will cancel out. Leave exit e m. There we go. So now only to do is take the integral of X idiom again, which we end up with on this right hand side exit the M plus one over AM plus one times, Ellen of X minus X NTM into grows in the xdm plus one. It's gonna be over exit a template one again. So you put this squared and then finally at a constant of plus C. So now we doesn't divide both sides by Xdm to get the final answer. So for the first round, we end up with X, the M plus one minus m. Because we're dividing by ex idiom over M plus one times Ellen of X, minus the same thing for the next one extra g M plus one minus Emma over M plus one squared and then plus C times extra d minus m because we divide by it. So now all we end up with is the M minus and get zero so x over m plus one Allah necks minus X over M plus one squared plus C to the exit of minus m.

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