Find the general solution for differential equations. (sin x) dy\dx} + (cos x) y = cos x sin^2 x

Question

Find the general solution for differential equations.

\((sin \, x)\frac{dy}{dx}\) + (cos x) y = sin²

(sin x) dy\dx} + (cos x) y = cos x sin² x

Answer 1

Given Differential Equation :

 \((sin \, x)\frac{dy}{dx}\) + (cos x) y = cos x. sin² x

Formula :

i) \(\int\) cot x dx = log(sin x)

ii) a^{log_a b} = b

iii) \(\int\) xⁿ dx = \(\frac{x^{n+1}}{n+1}\)

iv) General solution :

For the differential equation in the form of

\(\frac{dy}{dx}+\, Py\, = Q\)

General solution is given by,

y.(I. F.) = \(\int\) Q. (I.F.) dx + c

Where, integrating factor,

I.F. = \(e^{\int p\, dx}\)

Given differential equation is

sin x \(\frac{dy}{dx}\) + (cos x) y = cos x. sin² x

Dividing above equation by sin x,

Equation (1) is of the form

\(\frac{dy}{dx} \, + Py\, = Q\)

Where, \(P = cot\, x \, and\, Q = \, sin\, x. cos\, x\)

Therefore, integrating factor is

General solution is

Put sin x=t => cos x.dx=dt

Substituting I in eq(2),

Therefore, general solution is