The solution of the differential equation e^-x(y + 1)dy + (cos²(x) + sin(2x))y dx = 0 subjected to the condition that y = 1 when x = 0 is

• y + log(y) + e^xcos²(x) = 2

• log(y + 1) + e^xcos²(x) = 1

• y + log(y) = e^xcos²(x)

• (y + 1) + e^xcos²(x) = 2

The differential equation of the family of straight lines whose slope is equal to y-intercept is

• $\left(\mathrm{x} + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = 0$

• $\left(\mathrm{x} + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} - 1}{\mathrm{y} - 1}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} + 1}{\mathrm{y} + 1}$

The order and degree of the differential equation ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^5\right]}^{\frac{1}{3}} = \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ are respectively

• 1, 5

• 2, 1

• 2, 5

• 2, 3

The differential equation $\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{x} = \mathrm{c}$ represents

• a family of hyperbolas

• a family of circles whose centres are on the y-axis

• a family of parabolas

• a family of circles whose centres are on the x-axis

The differential equation of the family of circles passing through the orign and having their centres on the x-axis is

• ${\mathrm{y}}^2 = {\mathrm{x}}^2 +\hspace{0.17em}2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}$

• ${\mathrm{y}}^2 = {\mathrm{x}}^2 -\hspace{0.17em}2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}$

• ${\mathrm{x}}^2 = {\mathrm{y}}^2 +\hspace{0.17em}\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}$

• ${\mathrm{x}}^2 = {\mathrm{y}}^2 +\hspace{0.17em}3\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}$

A population grows at the rate of 10% of the population per year. How long does it take for the population to double?

• 20log(2) yr

• 10log(2)yr

• 5log(2) yr

• None of the above

The order and degree of the differential equation y = $\frac{\mathrm{dp}}{\mathrm{dx}}\mathrm{x} = \sqrt{{\mathrm{a}}^2{\mathrm{p}}^2 +\hspace{0.17em}{\mathrm{b}}^2}$, where p = $\frac{\mathrm{dy}}{\mathrm{dx}}$ (here a and b are arbitrary constants) respectively are

• 2, 2

• 1, 1

• 1, 2

• 2, 1

598.

The general solution of the differential equation $2\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = 3$ is a family of

• hyperbolas

• parabolas

• straight lines

• circles

If m and n are degree and order of ${\left(1 + {\mathrm{y}}_1^2\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = {\mathrm{y}}_2$, then the value of $\frac{\mathrm{m} + \mathrm{n}}{\mathrm{m} - \mathrm{n}}$ is

• 3

• 4

• 5

• 2

The general solution of ${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 1 - {\mathrm{x}}^2 - {\mathrm{y}}^2 + {\mathrm{x}}^2{\mathrm{y}}^2$ is

• $2{\sin }^{-1}\left(\mathrm{y}\right) = \mathrm{x}\sqrt{1 - {\mathrm{x}}^2} + {\sin }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

• ${\cos }^{-1}\left(\mathrm{y}\right) = {\mathrm{xcos}}^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

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

• $2{\sin }^{-1}\left(\mathrm{y}\right) = \mathrm{x}\sqrt{1 - {\mathrm{y}}^2} + \mathrm{C}$