Solve the equation.
$$
\frac{d y}{d x}=\frac{x}{\sin (y)}-\frac{9}{\sin (y)}
$$

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

Question

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

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To solve the differential equation $\frac{dy}{dx} = \frac{x}{\sin(y)} - \frac{9}{\sin(y)}$, we can combine the terms on the right-hand side to get $\frac{dy}{dx} = \frac{x-9}{\sin(y)}$. ‖ ‖ step 2 ⋮ We can then separate variables and integrate both sides: $\sin(y) dy = (x-9) dx$. ‖ ‖ step 3 ⋮ Integrating both sides gives us $-\cos(y) = -\frac{x^2}{2} + 9x + C$. ‖ ‖ step 4 ⋮ Solving for $y$, we get $y = \arccos\left(-\frac{x^2}{2} + 9x + C\right)$. ‖ ***A*** ∻Key Concept∻ ⚹Separation of Variables and Integration⚹ ∻Explanation∻ ⚹To solve the given first-order differential equation, we separated the variables and integrated both sides to find the general solution.⚹◊What is the solution to the differential equation $\frac{dy}{dx}=\frac{x}{\sin(y)}-\frac{9}{\sin(y)}$?⍭ Generate me a similar question◊

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