Question

Determine the order and degree of the given differential equation: $[1+ (\frac{dy}{dx})^{3}]^{\frac{7}{3}}=7\cdotp\frac{d^{2}y}{dx^{2}}$

Answer 1

Solution:
➖➖➖

➡️Order of a DE: Order is calculated by checking higher order derivative of x. All DE have order.

➡️Degree of DE: Degree is the power of highest order derivative,when complete equation is free from radicals. Every differential equation does not have degree.

Here in the given DE

$[1+ (\frac{dy}{dx})^{3}]^{\frac{7}{3}}=7\cdotp\frac{d^{2}y}{dx^{2}}$

taking cube both sides

$[1+ (\frac{dy}{dx})^{3}]^{7}= ({7\cdotp\frac{d^{2}y}{dx^{2}}})^{3}$
As on solving it we know that the derivatives does not change,So

here we can see that highest order derivative is 2,

ie Order is 2.

and the complete equation is free from radicals now , on expanding LHS to its 7 power , higher order derivative have no effect on its power,so here power of second order derivative is 3,

So, Degree = 3