Solve the equation. (dy)/(dx)=(x)/(7cos(y)) Choose 1 answer: (A) y=(cos(x)+x sin(x))/(7cos^(2)(x))+C (B) y=arcsin((x^(2))/(14)+C) (C) y=(cos(x)+x sin(x)+C)/(7cos^(2)(x)) (D) y=arcsin((x^(2))/(14))+C

Recognize type of differential equation: Recognize the type of differential equation. This is a first-order separable differential equation, which means we can separate the variables y and x on different sides of the equation.Separate the variables: Separate the variables. To separate the variables, we multiply both sides by 7cos(y)dy and divide by xdx. 7cos(y)dy = (x)/(7cos(y)) * 7cos(y)dx 7cos(y)dy = xdxIntegrate both sides: Integrate both sides. We integrate the left side with respect to y and the right side with respect to x. ∫7cos(y)dy = ∫xdxPerform the integration: Perform the integration. The integral of 7cos(y) with respect to y is 7sin(y), and the integral of x with respect to x is (1/2)x^2. 7sin(y) = (1/2)x^2 + C, where C is the constant of integration.Solve for y: Solve for y. To solve for y, we take the inverse sine (arcsin) of both sides. y = arcsin((x^2)/(14) + C/7)Match with given options: Match the solution with the given options. The solution we found is y = arcsin((x^2)/(14) + C/7), which is not exactly in the form of any of the given options. However, we can adjust the constant C to absorb the 7 in the denominator, so we rewrite the solution as: y = arcsin((x^2)/(14) + C) This matches option (B).

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Solve the equation.

(dy)/(dx)=(x)/(7cos(y))
Choose 1 answer:
(A)
y=(cos(x)+x sin(x))/(7cos^(2)(x))+C
(B)
y=arcsin((x^(2))/(14)+C)
(C)
y=(cos(x)+x sin(x)+C)/(7cos^(2)(x))
(D)
y=arcsin((x^(2))/(14))+C

Solve the equation. $\newline$ $\frac{d y}{d x}=\frac{x}{7 \cos (y)}$ $\newline$ Choose $1$ answer: $\newline$ (A) $y=\frac{\cos (x)+x \sin (x)}{7 \cos ^{2}(x)}+C$ $\newline$ (B) $y=\arcsin \left(\frac{x^{2}}{14}+C\right)$ $\newline$ (C) $y=\frac{\cos (x)+x \sin (x)+C}{7 \cos ^{2}(x)}$ $\newline$ (D) $y=\arcsin \left(\frac{x^{2}}{14}\right)+C$

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Find derivatives of using multiple formulae

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Q. Solve the equation.

\newline

\frac{d y}{d x}=\frac{x}{7 \cos (y)}

\newline

Choose

1

answer:

\newline

(A)

y=\frac{\cos (x)+x \sin (x)}{7 \cos ^{2}(x)}+C

\newline

(B)

y=\arcsin \left(\frac{x^{2}}{14}+C\right)

\newline

(C)

y=\frac{\cos (x)+x \sin (x)+C}{7 \cos ^{2}(x)}

\newline

(D)

y=\arcsin \left(\frac{x^{2}}{14}\right)+C

Recognize type of differential equation: Recognize the type of differential equation. $\newline$ This is a first-order separable differential equation, which means we can separate the variables $y$ and $x$ on different sides of the equation.
Separate the variables: Separate the variables. $\newline$ To separate the variables, we multiply both sides by $7\cos(y)\,dy$ and divide by $x\,dx$ . $\newline$ $7\cos(y)\,dy = \frac{x}{7\cos(y)} \times 7\cos(y)\,dx$ $\newline$ $7\cos(y)\,dy = x\,dx$
Integrate both sides: Integrate both sides. $\newline$ We integrate the left side with respect to $y$ and the right side with respect to $x$ . $\newline$ $\int 7\cos(y)\,dy = \int x\,dx$
Perform the integration: Perform the integration. $\newline$ The integral of $7\cos(y)$ with respect to $y$ is $7\sin(y)$ , and the integral of $x$ with respect to $x$ is $(1/2)x^2$ . $\newline$ $7\sin(y) = (1/2)x^2 + C$ , where $C$ is the constant of integration.
Solve for y: Solve for y. $\newline$ To solve for y, we take the inverse sine (arcsin) of both sides. $\newline$ $y = \arcsin\left(\frac{x^2}{14} + \frac{C}{7}\right)$
Match with given options: Match the solution with the given options. $\newline$ The solution we found is $y = \arcsin\left(\frac{x^2}{14} + \frac{C}{7}\right)$ , which is not exactly in the form of any of the given options. However, we can adjust the constant $C$ to absorb the $7$ in the denominator, so we rewrite the solution as: $\newline$ $y = \arcsin\left(\frac{x^2}{14} + C\right)$ $\newline$ This matches option (B).