Find the general solution for dy\dx​ + 2y cot x =3x^2 cosec^2x ​

Question

Find the general solution for \(\frac{dy}{dx} \) + 2y cot x =3x² cosec²x differential equations.

dy\dx + 2y cot x =3x² cosec²x

Answer 1

Given Differential Equation :

\(\frac{dy}{dx}\)+ 2y cot x =3x² cosec²x

Formula :

i) \(\int\) cot x dx = (sin x)

ii) alog b = log b^a 

iii) a^log_a ^b = b

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

v) 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

\(\frac{dy}{dx}\) 2y (cot x) = 3x² cosec²x …eq(1)

Equation (1) is of the form

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

Where, P = 2 cot x and Q = 3x² cosec²x

Therefore, integrating factor is

General solution is

Therefore, general solution is

y.(sin²x) = x³ + c