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Solve the given differential equation.$$\frac{d y}{d x}=\frac{(x+2) y^{3}}{x^{4}\left(y^{4}+3\right)}$$

$$3 x^{3} y^{4}-9 x^{3}=-3 x y^{2}-4 y^{2}+c x^{3} y^{2}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 2

Applications of Antidifferentiation

Integrals

Campbell University

University of Michigan - Ann Arbor

Idaho State University

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.

01:02

Solve the given differenti…

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03:20

Uh huh. We want to solve the given differential equation. Dy dx is equal to two X. Cubed over Y to the power of four. There are multiple methods to solve such a differential equation. Six dy and dx or rather sense of variables Y. And extremely isolated on either side of the problem easily. Or rather either side of the equation. The method we're going to solve its called separation of variables. So the first step is to isolate ry index terms on either side of the equation. This gives Y to the fourth. D. Y equals two X cubed dx. Next we want to integrate both sides to remove our differentials. Best integral Y to the fourth. Dy equals integral to the X. 30 X. This gives water the fifth over five equals one half X to the fourth plus C. However we want our equation in terms of why. So we need to isolate why. That's what we multiply both sides by five and take the fifth route we can obtain our final solution. This is why is equal to the 5th root A file over to extra 4th plus the constant of integration seat

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