31.

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{y\varphi }\text{'}\left(\mathrm{x}\right) - {\mathrm{y}}^2}{\mathrm{\varphi }\left(\mathrm{x}\right)} \mathrm{is} :$

• $\mathrm{y} = \frac{\mathrm{x}}{\mathrm{\varphi }\left(\mathrm{x}\right) + \mathrm{c}}$

• $\mathrm{y} = \frac{\mathrm{\varphi }\left(\mathrm{x}\right)}{ \mathrm{x}} + \mathrm{c}$

• $\mathrm{y} = \frac{\mathrm{\varphi }\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}}{\mathrm{x}}$

• $\mathrm{y} = \frac{\mathrm{\varphi }\left(\mathrm{x}\right)}{\mathrm{x} + \mathrm{c}}$

Let f(x) and g(x) be twice differentiable functions on [0, 2) satisfying f"(x) = g"(x), f'(1) = 4, g'(1) = 6,f(2) = 3 and g(2) = 9. Then what is f(x) - g(x) at x = 4 equal to

• - 10

•  - 6

•  - 4

• 2

What are the order and degree respectively of the differential equation whose solution is $\mathrm{y} = \mathrm{cx} + {\mathrm{c}}^2 - 3{\mathrm{c}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + 2$ where c is a parameter ?

• 1, 2

• 2, 2

• 1, 3

• 1, 4

$\mathrm{If} {\log }_{\mathrm{a}}\left(\mathrm{ab}\right) = \mathrm{x}, \mathrm{then} \mathrm{what} \mathrm{is} {\log }_{\mathrm{b}}\left(\mathrm{ab}\right) = ?$

• $\frac{1}{\mathrm{x}}$

• $\frac{\mathrm{x}}{\mathrm{x} + 1}$

• $\frac{\mathrm{x}}{1 - \mathrm{x}}$

• $\frac{\mathrm{x}}{\mathrm{x} - 1}$

If $\mathrm{y} = {\log }_{10}\left(\mathrm{x}\right) + {\log }_{\mathrm{x}}\left(10\right) + {\log }_{\mathrm{x}}\left(\mathrm{x}\right) + {\log }_{10}\left(10\right)$ then what is

${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\mathrm{x} = 10} = ?$

• 10

• 2

• 1

• 0

What are the degree and order respectively of the differential equation satisfying

${\mathrm{e}}^{\mathrm{y}\sqrt{1 - {\mathrm{x}}^2} + \mathrm{x}\sqrt{1 - {\mathrm{y}}^2}} = {\mathrm{ce}}^{\mathrm{x}}, \mathrm{where} \left(\mathrm{c} > 0, \left|\mathrm{x}\right| < 1, \left|\mathrm{y}\right| < 1\right) ?$

• 1, 1

• 1, 2

• 2, 1

• 2, 2

If xdy = ydx + ydy, y > 0 and y(1) = 1, then what is y( - 3) equal to ?

• 3 only

•  - 1 only

• Both - 1 and 3

• Neither  - 1 nor 3

What is the order of the differential equation $\frac{\mathrm{dx}}{\mathrm{dy}} + \int \mathrm{ydx} = {\mathrm{x}}^3 ?$

• 1

• 2

• 3

• Cannot be determined

Which one of the following differential equation is represents the family of straight lines which are at unit distance from the origin ?

• ${\left(\mathrm{y} - \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 1 - {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$

• ${\left(\mathrm{y} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$

• ${\left(\mathrm{y} - \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$

• ${\left(\mathrm{y} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 1 - {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$

What is the solution of x ≤ 4, y ≥ 0 and x ≤ - 4, y ≤ 0 ?

• x ≥ - 4, y ≤ 0

• x ≤ 4, y ≥ 0

• x ≤ - 4, y = 0

• x ≥ - 4, y = 0