Find the order and degree of the following differential equations. i) `(dy)/(dx)+y=1/((dy)/(dx))`, ii) `e^(e^(3)y)/(dx^(3))-x(d^(2)y)/(dx^(2))+y=0` , iii) `sin^(-1)(dy)/(dx)=x+y`, iv) `log_(e)(dy)/(dx)=ax+by`
v) `y(d^(2)y)/(dx^(2))+x((dy)/(dx))^(2)-4y(dy)/(dx)=0`

Explanation

i) `(dy)/(dx) +y=1/(dy)/(dx)`
`therefore ((dy)/(dx))^(2)+y(dy)/(dx)=1`
Hence order is 1 and degree is 2.
ii) `e^(d^(3)y)/(dx^(3))-x(d^(2)y)/(dx^(2))+y=0`
Clearly order is 3, but degree is not defined as it cannot be expressed as a polynomial equation in derivatives.
iii) `sin^(-1)(dy)/(dx)=x+y`
`therefore (dy)/(dx)=sin(x+y)`
Hence order is 1 and degree is 1.
iv) `log_(e)(dy)/(dx)=ax+by`
`therefore (dy)/(dx)=e^(ax+by)`
Hence order is one and degree is also 1.
v) `y(d^(2)y)/(dx^(2))+x((dy)/(dx))^(2)-4y(dy)/(dx)=0`
Clearly, order is 2 and degree is 1.