The integrating factor of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = 2{\mathrm{x}}^2$ is

• $\frac{1}{\mathrm{x}}$

• x

• e^-x

• e^-y

If $\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}\left(\mathrm{x}\right) = 4{\mathrm{x}}^3 - \frac{3}{{\mathrm{x}}^4}$ such that f(2) = 0. Then, f(x) is

• ${\mathrm{x}}^3 + \frac{1}{{\mathrm{x}}^4} - \frac{129}{8}$

• ${\mathrm{x}}^4 + \frac{1}{{\mathrm{x}}^3} + \frac{129}{8}$

• ${\mathrm{x}}^3 + \frac{1}{{\mathrm{x}}^4} + \frac{129}{8}$

• ${\mathrm{x}}^4 + \frac{1}{{\mathrm{x}}^3} - \frac{129}{8}$

The order and degree of the differential equation ${\left(\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}\right)}^2 - 3\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 2{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^4 = {\mathrm{y}}^4$ are

• 1, 4

• 3, 4

• 2, 4

• 3, 2

The solution of the differential equation (1 + y²) dx = (tan^-1((y) - x)dy is

• ${\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{y}\right)} = (1 - {\tan }^{-1}\left(\mathrm{y}\right)){\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{C}$

• ${\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{y}\right)} = ({\tan }^{-1}\left(\mathrm{y}\right) - 1){\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{C}$

• $\mathrm{x} = {\tan }^{-1}\left(\mathrm{y}\right) - 1 + {\mathrm{Ce}}^{{\tan }^{-1}\left(\mathrm{y}\right)}$

• None of the above

The solution of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y} - \mathrm{xtan}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)$ is

• $\mathrm{xsin}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \mathrm{C} = 0$

• $\mathrm{xsin}\left(\mathrm{y}\right) + \mathrm{C} = 0$

• $\mathrm{xsin}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{C}$

• None of these

The particular solution of $\cos \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \mathrm{a}\left(\mathrm{where}, \mathrm{a} \in \mathrm{R}\right)$, (y = 2 when x = 0), is

• $\cos \left(\frac{\mathrm{y} - 2}{\mathrm{x}}\right) = \mathrm{a}$

• $\sin \left(\frac{\mathrm{y} - 2}{\mathrm{x}}\right) = \mathrm{a}$

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

• $\mathrm{y} = {\mathrm{acos}}^{-1}\left(\mathrm{x}\right)$

The order and degree of the differential equation $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = {\left\{\mathrm{y} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right\}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ are given by

• 4 and 2

• 1 and 2

• 1 and 4

• 2 and 4

318.

The differential equation of the family of circles touching the y-axis at the origin is

• xy' - 2y = 0

• y'' - 4y' + 4y = 0

• 2xyy' + x² = y²

• 2yy' + y² = x²

Solution of the equation ${\cos }^2\left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}} - \left(\tan \left(2\mathrm{x}\right)\right)\mathrm{y} = {\cos }^2\left(\mathrm{x}\right), \left|\mathrm{x}\right| < \frac{\mathrm{\pi }}{4}$, where $\mathrm{y}\left(\frac{\mathrm{\pi }}{6}\right) = \frac{3\sqrt{3}}{8}$, is given by 

• $\mathrm{y}\frac{\tan \left(2\mathrm{x}\right)}{1 - {\tan }^2\left(\mathrm{x}\right)} = 0$

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

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

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

The equation of the curve through the point (1, 0), whose slope is $\frac{\mathrm{y} - 1}{{\mathrm{x}}^2 + \mathrm{x}}$, is

• 2x(y - 1) + x + 1 = 0

• (x + 1)(y - 1) + 2x = 0

• x(y - 1)(x + 1) + 2 = 0

• None of these