Write degree of the differential equation d 2 y d x 2 + 3 d y d x 2 = x 2 log d 2 y d x 2

Given differential equation is d 2 y d x 2 + 3 d y d x 2 = x 2 log d 2 y d x 2 Here d 2 y d x 2 is in log Therefore, the given differential equation is not a polynomial equation in derivatives. Hence, the degree for this differential equation is undefined.

Write the degree of the differential equation d 2 y d x 2 2 + d y d x 2 = x sin d y d x

Given differential equation is d 2 y d x 2 2 + d y d x 2 = x sin d y d x Here d y d x is in the sin . Therefore, the given differential equation is not a polynomial equation in derivatives. Hence, the degree for this differential equation is not defined.

Write the order and degree of the differential equation d 2 y d x 2 + d y d x 1 4 4 + x 1 5 = 0

Given differential equation is d 2 y d x 2 + d y d x 1 4 4 + x 1 5 = 0 Highest order derivative in the differential equation is d 2 y d x 2 . Therefore, the order of the differential equation is 2 . Here, d y d x has a fractional power. The given differential equation is not a polynomial equation in derivatives. Hence, the degree for this differential equation is not defined.

The degree of the differential equation d 2 y d x 2 + e d y d x = 0

Given differential equation is d 2 y d x 2 + e d y d x = 0 . Here d y d x is in the power of e . Therefore, given differential equation is not a polynomial equation in derivatives. Hence, the degree for this differential equation is undefined.

Find the solution of the differential equation x 1 + y 2 d x + y 1 + x 2 d y - 0

Given differential equation is x 1 + y 2 d x + y 1 + x 2 d y = 0 ⇒ y 1 + x 2 d y = - x 1 + y 2 d x ⇒ y 1 + y 2 d y = - x 1 + x 2 d x This is a variable separable form of differential equation. ⇒ ∫ y 1 + y 2 d y = - ∫ x 1 + x 2 d x .........(i) Let 1 + y 2 = t 2 and 1 + x 2 = p 2 ⇒ 2 y d y = 2 t d t and 2 x d x = 2 p d p ⇒ y d y = t d t and x d x = p d p From (i), we get ∫ d t = - ∫ d p ⇒ t = - p + C ⇒ t + p = C ∴ 1 + y 2 + 1 + x 2 = C is the solution of the given differential equation.