Solve the equation. (dy)/(dx)=(x)/(sin(y))-(9)/(sin(y)) Choose 1 answer: (A) y=arccos(-(x^(2))/(2)+9x+C) (B) y=arccos(-(x^(2))/(2)+9x)+C (C) y=(2)/(cos(-x^(2)+18 x+C)) (D) y=(2C)/(cos(-x^(2)+18 x))

Simplify the equation: First, we need to simplify the right-hand side of the differential equation by combining the terms over a common denominator. (dy)/(dx) = (x)/(sin(y)) - (9)/(sin(y)) (dy)/(dx) = (x - 9)/(sin(y))Separate variables: Next, we separate the variables x and y to different sides of the equation to prepare for integration. sin(y) dy = (x - 9) dxIntegrate both sides: Now, we integrate both sides of the equation with respect to their respective variables. ∫sin(y) dy = ∫(x - 9) dxSolve for y: The integral of sin(y) with respect to y is -cos(y), and the integral of (x - 9) with respect to x is (x^2)/2 - 9x. -cos(y) = (x^2)/2 - 9x + C, where C is the constant of integration.Check solution: We solve for y by taking the arccosine of both sides. y = arccos(-((x^2)/2 - 9x + C))Final answer: We need to check if the solution matches any of the given answer choices. The correct answer choice that matches our solution is: (B) y = arccos(-(x^2)/2 + 9x) + C

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Solve the equation.

(dy)/(dx)=(x)/(sin(y))-(9)/(sin(y))
Choose 1 answer:
(A)
y=arccos(-(x^(2))/(2)+9x+C)
(B)
y=arccos(-(x^(2))/(2)+9x)+C
(C)
y=(2)/(cos(-x^(2)+18 x+C))
(D)
y=(2C)/(cos(-x^(2)+18 x))

Solve the equation. $\newline$ $\frac{d y}{d x}=\frac{x}{\sin (y)}-\frac{9}{\sin (y)}$ $\newline$ Choose $1$ answer: $\newline$ (A) $y=\arccos \left(-\frac{x^{2}}{2}+9 x+C\right)$ $\newline$ (B) $y=\arccos \left(-\frac{x^{2}}{2}+9 x\right)+C$ $\newline$ (C) $y=\frac{2}{\cos \left(-x^{2}+18 x+C\right)}$ $\newline$ (D) $y=\frac{2 C}{\cos \left(-x^{2}+18 x\right)}$

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Find derivatives of using multiple formulae

Full solution

Q. Solve the equation.

\newline

\frac{d y}{d x}=\frac{x}{\sin (y)}-\frac{9}{\sin (y)}

\newline

Choose

1

answer:

\newline

(A)

y=\arccos \left(-\frac{x^{2}}{2}+9 x+C\right)

\newline

(B)

y=\arccos \left(-\frac{x^{2}}{2}+9 x\right)+C

\newline

(C)

y=\frac{2}{\cos \left(-x^{2}+18 x+C\right)}

\newline

(D)

y=\frac{2 C}{\cos \left(-x^{2}+18 x\right)}

Simplify the equation: First, we need to simplify the right-hand side of the differential equation by combining the terms over a common denominator. $\newline$ $\frac{dy}{dx} = \frac{x}{\sin(y)} - \frac{9}{\sin(y)}$ $\newline$ $\frac{dy}{dx} = \frac{x - 9}{\sin(y)}$
Separate variables: Next, we separate the variables $x$ and $y$ to different sides of the equation to prepare for integration. $\sin(y) \, dy = (x - 9) \, dx$
Integrate both sides: Now, we integrate both sides of the equation with respect to their respective variables. $\newline$ $\int \sin(y) \, dy = \int (x - 9) \, dx$
Solve for $y$ : The integral of $\sin(y)$ with respect to $y$ is $-\cos(y)$ , and the integral of $(x - 9)$ with respect to $x$ is $\frac{x^2}{2} - 9x$ . $\newline$ $-\cos(y) = \frac{x^2}{2} - 9x + C$ , where $C$ is the constant of integration.
Check solution: We solve for $y$ by taking the arccosine of both sides. $y = \arccos\left(-\left(\frac{x^2}{2} - 9x + C\right)\right)$
Final answer: We need to check if the solution matches any of the given answer choices. $\newline$ The correct answer choice that matches our solution is: $\newline$ (B) $y = \arccos\left(-\frac{x^2}{2} + 9x\right) + C$