Find the general solution of the differential equation $$\sin(x)\dfrac{dy}{dx} + \cos(x)y = 2\cos(x)$$

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The given differential equation is sin(x) dy/dx + cos(x) y = 2 cos(x).
First, divide through by sin(x) (where sin(x) ≠ 0) to obtain the standard linear form dy/dx + cot(x) y = 2 cot(x).
This is a first-order linear ODE with integrating factor μ(x) = exp(∫ cot(x) dx) = exp(ln|sin(x)|) = sin(x).
Multiply the entire equation by μ(x): sin(x) dy/dx + cos(x) y = 2 cos(x) becomes d/dx [sin(x) y] = 2 cos(x).
Integrate both sides with respect to x: ∫ d/dx [sin(x) y] dx = ∫ 2 cos(x) dx, which yields sin(x) y = 2 sin(x) + C, where C is a constant of integration.
Solve for y: y = 2 + C / sin(x) = 2 + C csc(x).
Thus the general solution can be written as y(x) = 2 + A / sin(x) for an arbitrary constant A.
If one rewrites A as C, this matches the form y(x) = A csc(x) + 2. Note that the original equation has potential issues at sin(x) = 0, where the division step is not valid, but in the regions where sin(x) ≠ 0, the above expression describes all solutions.

ENG1090 - MUM S1 2025 Mock Final Exam

Find the general solution of the differential equation $$\sin(x)\dfrac{dy}{dx} + \cos(x)y = 2\cos(x)$$

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