51.

If xdy = y(dx + y dy), y(1) = 1 and y(x) > 0, then y(- 3) is equal to

• 3

• 2

• 1

• 0

If ${\int }_{\mathrm{a}}^{\mathrm{b}}{\mathrm{x}}^3\mathrm{dx} = 0 \mathrm{and} {\int }_{\mathrm{a}}^{\mathrm{b}}{\mathrm{x}}^2\mathrm{dx} = \frac{2}{3}$, then the values of a and b are respectively

• 1, - 1

• - 1, 1

• 1, 1

• - 1, - 1

The differential equation representing the family of curves y = 2c (x + c³), where c is a positive parameter, is of

• order 1, degree 1

• order 1, degree 2

• order 1, degree 3

• order 1, degree 4

The differential equation representing the family of curves y = xe^cx (c is a constant) is

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

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

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

• $\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}1 = \frac{\mathrm{y}}{\mathrm{x}}\left(\log \left(\frac{\mathrm{y}}{\mathrm{x}}\right)\right)$

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} = 1 + \mathrm{y} + {\mathrm{y}}^2 + \mathrm{x} + \mathrm{xy} + {\mathrm{xy}}^2$ is

• ${\tan }^{-1}\left(\frac{2\mathrm{y} + 1}{\sqrt{3}}\right) = \mathrm{x} + \mathrm{xy} + {\mathrm{xy}}^2$

• $4{\tan }^{-1}\left(\frac{2\mathrm{y} + 1}{\sqrt{3}}\right) = \frac{\sqrt{3}}{2}\left(2\mathrm{x} + {\mathrm{x}}^2\right)$

• $\sqrt{3}{\tan }^{-1}\left(\frac{3\mathrm{y} + 1}{\sqrt{3}}\right) = 4\left(1 +\hspace{0.17em}\mathrm{x} +\hspace{0.17em}{\mathrm{x}}^2\right) + \mathrm{c}$

• ${\tan }^{-1}\left(\frac{2\mathrm{y} + 1}{\sqrt{3}}\right) = \sqrt{3}\left(2\mathrm{x} + {\mathrm{x}}^2\right) + \mathrm{c}$

The integrating factor of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{y}}{\left(1 - \mathrm{x}\right){\displaystyle \sqrt{\mathrm{x}}}} = 1 - \sqrt{\mathrm{x}}$ is

• $\frac{1 - \sqrt{\mathrm{x}}}{1 + \sqrt{\mathrm{x}}}$

• $\frac{1 + \sqrt{\mathrm{x}}}{1 - \sqrt{\mathrm{x}}}$

• $\frac{1 - \mathrm{x}}{1 +\hspace{0.17em}\mathrm{x}}$

• $\frac{\sqrt{\mathrm{x}}}{1 - \sqrt{\mathrm{x}}}$