What are the degree and order respectively of the differential equation

$\mathrm{y} = \mathrm{x}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 + {\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)}^2 ?$

• 1, 2

• 2, 1

• 1, 4

• 4, 1

What is the differential equation corresponding to y² - 2ay + x² = a² by eliminating a² ? where p = $\frac{\mathrm{dy}}{\mathrm{dx}}$

• $\left({\mathrm{x}}^2 - 2{\mathrm{y}}^2\right){\mathrm{p}}^2 - 4\mathrm{pxy} - {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 - 2{\mathrm{y}}^2\right){\mathrm{p}}^2 + 4\mathrm{pxy} - {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 + 2{\mathrm{y}}^2\right){\mathrm{p}}^2 - 4\mathrm{pxy} - {\mathrm{x}}^2 = 0$

• $\left({\mathrm{x}}^2 + 2{\mathrm{y}}^2\right){\mathrm{p}}^2 - 4\mathrm{pxy} + {\mathrm{x}}^2 = 0$

23.

What is the general solution to the differential equation

ydx - (x + 2y²)dy = 0 ? 

• x = y² + cy

• x = 2cy²

• x = 2y² + cy

• None of the above

Let f(x + y) = f(0)f(y) for all x and y. Then what is f'(5) equal to [where f'(x) is the derivative of f(x)] ?

• $\mathrm{f}\left(5\right)\mathrm{f}\text{'}\left(0\right)$

• $\mathrm{f}\left(5\right) - \mathrm{f}\text{'}\left(0\right)$

• $\mathrm{f}\left(5\right)\mathrm{f}\left(0\right)$

• $\mathrm{f}\left(5\right) +\hspace{0.17em}\mathrm{f}\text{'}\left(0\right)$

If $\mathrm{y} = {\left(\cos \left(\mathrm{x}\right)\right)}^{{\left(\cos \left(\mathrm{x}\right)\right)}^{\left(\cos \left(\mathrm{x}\right)\right)}}, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $- \frac{{\mathrm{y}}^2\tan \left(\mathrm{x}\right)}{1 - \mathrm{yln}\left(\cos \left(\mathrm{x}\right)\right)}$

• $\frac{{\mathrm{y}}^2\tan \left(\mathrm{x}\right)}{1 + \mathrm{yln}\left(\cos \left(\mathrm{x}\right)\right)}$

• $\frac{{\mathrm{y}}^2\tan \left(\mathrm{x}\right)}{1 - \mathrm{yln}\left(\sin \left(\mathrm{x}\right)\right)}$

• $\frac{{\mathrm{y}}^2\tan \left(\mathrm{x}\right)}{1 + \mathrm{yln}\left(\sin \left(\mathrm{x}\right)\right)}$

$$\begin{gathered} \mathrm{If} {\mathrm{l}}_1 = \frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\sin \left(\mathrm{x}\right)}\right) \\ {\mathrm{l}}_2 = \underset{\mathrm{h}\to 0}{\lim }\frac{{\mathrm{e}}^{\sin \left(\mathrm{x} + \mathrm{h}\right)} - {\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}{\mathrm{h}} \\ {\mathrm{l}}_3 = \int {\mathrm{e}}^{\sin \left(\mathrm{x}\right)}\cos \left(\mathrm{x}\right)\mathrm{dx} \\ \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• ${\mathrm{l}}_1 \ne {\mathrm{l}}_2$

• $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{l}}_3\right) = {\mathrm{l}}_2$

• $\int {\mathrm{l}}_3\mathrm{dx} = {\mathrm{l}}_2$

• l₂ = l₃

The general solution of

$$\begin{gathered} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{ax} + \mathrm{h}}{\mathrm{by} + \mathrm{k}} \\ \mathrm{represents} \mathrm{a} \mathrm{circle} \mathrm{only} \mathrm{when} \end{gathered}$$

• a = b = 0

• $\mathrm{a} = - \mathrm{b} \ne 0$

• $\mathrm{a} = \mathrm{b} \ne 0, \mathrm{h} = \mathrm{k}$

• $\mathrm{a} = \mathrm{b} \ne 0$

The order and degree of the differential equation

${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^3 = {\mathrm{\rho }}^2{\left[\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right]}^2$ are respectively :

• 3 and 2

• 2 and 2

• 2 and 3

• 1 and 3

If y = ${\cos }^{-1}\left(\frac{2\mathrm{x}}{1 + {\mathrm{x}}^2}\right), \mathrm{Then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $- \frac{2}{1 + {\mathrm{x}}^2} \mathrm{for} \mathrm{all} \left|\mathrm{x}\right| < 1$

• $- \frac{2}{1 + {\mathrm{x}}^2} \mathrm{for} \mathrm{all} \left|\mathrm{x}\right| > 1$

• $\frac{2}{1 + {\mathrm{x}}^2} \mathrm{for} \mathrm{all} \left|\mathrm{x}\right| < 1$

• None of the above

The differential equation of minimum order by eliminating the arbitrary constants A and C in the equation y = A[sin (x + C) + cos(x + C)] is :

• y'' + (sin(x) + cos(x))y' = 1

• y'' = (sin(x) + cos(x))y'

• y'' = (y')² + (sin(x)cos(x))

• y'' + y = 0