Solve the differential equation. {eq}\cos^2 x \sin x \frac{dy}{dx} + y \cos^3 x = 1 {/eq}

Question:

Solve the differential equation.

{eq}\cos^2 x \sin x \frac{dy}{dx} + y \cos^3 x = 1 {/eq}

Differential Equation Using Integrating Factor:

Here we have used Integrating factor to solve the given differential equation.

To find the integrating factor w have used the following formula:

{eq}I.F.= e^{\int \cot x}dx\\=e^{ln(\sin x)}\\=\sin x {/eq}

We have also used the following formula to further solve the given equation:

{eq}\int \sec^{2} x=\tan x+c {/eq}

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The given equation is:

{eq}(\cos^{2} x\sin x)\frac{\mathrm{d} y}{\mathrm{d} x}+y\cos^{3} x =1 {/eq}

On Solving we get:

{eq}\dfrac{\mathrm{d}...