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3 years ago

Simplify 4(xy^2)^3 x (2x^3y)^2

Mathematics

1 answer:

tangare [24]3 years ago

7 0

4(xy²)³ · (2x³y)²
4(x³y⁶) · 4x⁶y²
4x³y⁶ · 4x⁶y²
16x⁹y⁸

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Show that an implicit solution of 2x sin2(y) dx - (x2 + 11) cos(y) dy = 0 is given by ln(x2 + 11) + csc(y) = C.

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From the question we are told that

The equation is $2x sin^2(y) dx - (x^2 + 11) cos(y) dy = 0$

Generally the goal of this solution is to prove that the implicit solution of the differential equation above is $ln(x^2 + 11) + csc(y) = C.$

First Step

Differentiate the solution with respect to x

$0 = \frac{2x}{ x^2 + 11} + [-cot * csc(y) ]\frac{dy}{dx}$

=> $0 = \frac{2x}{ x^2 + 11} + [-\frac{cos(y)}{ sin(y) }* \frac{1}{sin(y)} ]\frac{dy}{dx}$

=> $0 = \frac{2x}{ x^2 + 11} + [-\frac{cos(y)}{ sin^2(y) }]\frac{dy}{dx}$

multiply through by $sin^2(y)$ , $x^2 + 11$ , $dx$

=> $2 x (sin^2(y) dx + [-cos(y) (x^2 + 11)]dy = 0$

Looking at the equation obtained we see that it is equivalent to the differential equation hence the implicit solution of $2x sin^2(y) dx - (x^2 + 11) cos(y) dy = 0$ is $ln(x^2 + 11) + csc(y) = C.$

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