451.

The order and degree of the differential equation

${\left(1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right)}^{\frac{3}{4}} = {\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^{\frac{1}{3}}$

• (2, 4)

• (2, 3)

• (6, 4)

• (6, 9)

The integrating factor of the differential equation $\left(\mathrm{ylog}\left(\mathrm{y}\right)\right)\mathrm{dx} = \left(\log \left(\mathrm{y}\right) - \mathrm{x}\right)\mathrm{dy}$ is

• $\frac{1}{\log \left(\mathrm{y}\right)}$

• $\log \left(\log \left(\mathrm{y}\right)\right)$

• $1 + \log \left(\mathrm{y}\right)$

• $\log \left(\mathrm{y}\right)$

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

• y = - x² - 2x - 2 + ce^x

• y = x² + 2x + 2 - ce^x

• x = - y² - 2y + 2 - ce^y

• x = - y² - 2y - 2 + ce^y

The solution of the equation ${\left(3 + 2\sqrt{2}\right)}^{{\mathrm{x}}^{2 }- 8} + {\left(3 + 2\sqrt{2}\right)}^{8 - {\mathrm{x}}^2}$ = 6 are

• $3 \pm 2\sqrt{2}$

• $\pm 1$

• $\pm 3\sqrt{3}, \pm 2\sqrt{2}$

• $\pm 3, \pm \sqrt{7}$

The family of curves y = e^asin(x), where a is anarbitrary constant, is represented by thedifferential equation 

• $\log \left(\mathrm{y}\right) = \tan \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}}$

• $\mathrm{ylog}\left(\mathrm{y}\right) = \tan \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}}$

• $\mathrm{ylog}\left(\mathrm{y}\right) = \sin \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}}$

• $\log \left(\mathrm{y}\right) = \cos \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}}$

The integrating factor of $\mathrm{x} \frac{d\mathrm{y}}{d\mathrm{x}} + \left(1 + \mathrm{x}\right) \mathrm{y} = \mathrm{x}$ is

• x

• 2x

• e^xlog(x)

• xe^x

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + 1 = {\mathrm{e}}^{\mathrm{x} + \mathrm{y}}$ is

• x + e^{x + y} = c

• x - e^{x + y} = c

• x + e^{- (x + y)} = c

• x - e^{- (x + y)} = c

The degree and order of the differential equation $\mathrm{y} = \mathrm{px} + \sqrt[3]{{\mathrm{a}}^2{\mathrm{p}}^2 + {\mathrm{b}}^2},$ where p = $\frac{\mathrm{dy}}{\mathrm{dx}}$, are respectively.

• 3, 1

• 1, 3

• 1, 1

• 3, 3

The differential equation representing the family of curves y² = 2c (x + $\sqrt{\mathrm{c}}$) where c is a positive parameter, is of

• order 1, degree 2

• order 1, degree 3

• order 2, degree 3

• order 2, degree 2

An integrating factor of the differential equation $\left(1 + {\mathrm{x}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = \mathrm{x}$ is

• $\frac{\mathrm{x}}{1 + {\mathrm{x}}^2}$

• $\frac{1}{2}\log \left(1 + {\mathrm{x}}^2\right)$

• $\sqrt{1 + {\mathrm{x}}^2}$

• x