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

• y = e^x + C

• y = x + e^x + C

• y = xe^x + C

• y = x(e^x + 1) + C

The order and degree of the differential equation $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ = y are respectively

• 1, 1

• 1, 2

• 1, 3

• 2, 2

133.

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

• sin²(x)

• $\frac{2}{\sin \left(\mathrm{x}\right)}$

• $\log \left|\sin \left(\mathrm{x}\right)\right|$

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

If x² + y² = 1, then

• yy'' + (y')² + 1 = 0

• yy'' +2 (y')² + 1 = 0

• yy'' - 2(y')² + 1 = 0

• yy'' + (y')² - 1 = 0

The solution of the differential equation y'(y² - x) = y is

• y³ - 3xy = C

• y³ + 3xy = C

• x³ - 3xy = C

• y³ - xy = C

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

• 2 and 2

• 2 and 1

• 3 and 2

• 3 and 3

The slope of a curve at any point (x, y) other than the origin, is y + $\frac{\mathrm{y}}{\mathrm{x}}$. Then, the equation of the curve is

• y = C xe^x

• y = x(e^x + C)

• xy = Ce^x

• y + xe^x = C

The general solution of the differential equation $\left(\mathrm{x} + \mathrm{y} + 3\right)\frac{\mathrm{dy}}{\mathrm{dx}} = 1$ is

• x + y + 3 = Ce^y

• x + y + 4 = Ce^y

• x + y + 3 = Ce^- y

• x + y + 4 = Ce^- y

The differential equation representing the family of curves y² = a(ax + b), where a and b are arbitrary constants, is of

• order 1, degree 1

• order 1, degree 3

• order 2, degree 3

• order 2, degree 1

The solution of the differential equation $\frac{\mathrm{x}{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}} - \mathrm{y}}{\sqrt{{\mathrm{x}}^2 - {\mathrm{y}}^2}} = 10{\mathrm{x}}^2$ is

• ${\sin }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) - 5{\mathrm{x}}^2$ = C

• ${\sin }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = 10{\mathrm{x}}^2$ + C

• $\frac{\mathrm{y}}{\mathrm{x}} = 5{\mathrm{x}}^2 + \mathrm{C}$

• ${\sin }^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) = 10{\mathrm{x}}^2 + \mathrm{Cx}$