Find order and degree of the following differential equations.

$\text{ i) }\frac{dy}{dx}=\frac{x^{1/2}}{y^{1/2}(1+x{)}^{1/2}}$                                       $\text{ ii) }\frac{d^{2}y}{d{x}^2}={\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^{5/3}$

$\text{ iii) }1+{\left[\frac{d^{2}y}{d{x}^2}\right]}^2={\left[2+{\left(\frac{dy}{dx}\right)}^2\right]}^{3/2}$                 $\text{ iv) }\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+y=\log \left(\frac{dy}{dx}\right)$

$\text{ v) }{\left[{\left(\frac{dy}{dx}\right)}^{1/2}+{\left(\frac{d^{2}y}{d{x}^2}\right)}^{1/3}\right]}^{1/4}=0$                  $\text{ vi) }\frac{d^{2}y}{d{x}^2}=-p^{2}y$

$\text{ vii) }{\left[\frac{d^{3}y}{d{x}^3}\right]}^2-3{\left(\frac{dy}{dx}\right)}^2-e^x=4$                        $\text{ viii) }{x}^{1/2}{\left(\frac{d^{2}y}{d{x}^2}\right)}^{1/3}+x\frac{dy}{dx}+y=0$

$\text{ ix) }{\left[\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3\right]}^{6/5}=6y$

i) Highest derivative is $\frac{dy}{dx},$

then order is 1 highest derivative power is one $\text{ i.e }{\left(\frac{dy}{dx}\right)}^1\text{,}$ then degree is 1.

$\text{ ii) }\left(\frac{d^{2}y}{d{x}^2}\right)={\left[{\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^5\right]}^{1/3}$

cubing on both sides

${\left(\frac{d^{2}y}{d{x}^2}\right)}^3={\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^5$

Order -2, degree -3

$\text{ iii) }1+{\left(\frac{d^{2}y}{d{x}^2}\right)}^2={\left[{\left[2+{\left(\frac{dy}{dx}\right)}^2\right]}^3\right]}^{1/2}$

squaring on both sides

${\left[1+{\left(\frac{d^{2}y}{d{x}^2}\right)}^2\right]}^2={\left[2+{\left(\frac{dy}{dx}\right)}^2\right]}^3$

Order-2, degree-4

iv) Order -2 , degree not defined since the equation cannot be expressed as a polynomial equation in the derivatives.

$\text{ v) }{\left[{\left(\frac{dy}{dx}\right)}^{1/2}+{\left(\frac{d^{2}y}{d{x}^2}\right)}^{1/3}\right]}^{1/4}=0$

${\left[{\left[{\left(\frac{dy}{dx}\right)}^{1/2}+{\left(\frac{d^{2}y}{d{x}^2}\right)}^{1/3}\right]}^{1/4}\right]}^4=0$

${\left(\frac{dy}{dx}\right)}^{1/2}+{\left(\frac{d^{2}y}{d{x}^2}\right)}^{1/3}=0$

${\left[{\left(\frac{dy}{dx}\right)}^{1/2}\right]}^6={\left[{\left(\frac{d^{2}y}{d{x}^2}\right)}^{1/3}\right]}^6$

${\left(\frac{dy}{dx}\right)}^3={\left(\frac{d^{2}y}{d{x}^2}\right)}^2$

Order -2, degree -2
vi) Order -2, degree -1
vii) Order -3, degree -2
viii) Order -2, degree-1
ix) Order -2, degree -1