Mathematics : Differential Equations

Integrating factor of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}\mathrm{P}\left(\mathrm{x}\right)\mathrm{y} = \mathrm{Q}\left(\mathrm{x}\right)$ is :

• $\int \mathrm{P} \mathrm{dx}$

• $\int \mathrm{Q} \mathrm{dx}$

• ${\mathrm{e}}^{\int \mathrm{P} \mathrm{dx}}$

• ${\mathrm{e}}^{\int \mathrm{Q} \mathrm{dx}}$

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{1 + \cos \left(2\mathrm{y}\right)}{1 - \cos \left(2\mathrm{y}\right)} = 0$ is given by

• $\tan \left(\mathrm{y}\right) + \cot \left(\mathrm{x}\right) = \mathrm{c}$

• $\tan \left(\mathrm{y}\right) - \cot \left(\mathrm{x}\right) = \mathrm{c}$

• $\tan \left(\mathrm{x}\right) - \cot \left(\mathrm{y}\right) = \mathrm{c}$

• $\tan \left(\mathrm{x}\right) + \cot \left(\mathrm{y}\right) = \mathrm{c}$

The degree of the differential equation ${\left(1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} = {\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ is

• 2

• 4

• 9

• 1

The differential equation for which sin^-1(x) + sin^-1(y) = c is given by

• $\sqrt{1 - {\mathrm{x}}^2}\mathrm{dy} + \sqrt{1 - {\mathrm{y}}^2}\mathrm{dx} = 0$

• $\sqrt{1 - {\mathrm{x}}^2}\mathrm{dx} + \sqrt{1 - {\mathrm{y}}^2}\mathrm{dy} = 0$

• $\sqrt{1 - {\mathrm{x}}^2}\mathrm{dx} - \sqrt{1 - {\mathrm{y}}^2}\mathrm{dy} = 0$

• $\sqrt{1 - {\mathrm{x}}^2}\mathrm{dy} - \sqrt{1 - {\mathrm{y}}^2}\mathrm{dx} = 0$

The differential of ${\mathrm{e}}^{{\mathrm{x}}^3}$ with respect to log(x) is

• ${\mathrm{e}}^{{\mathrm{x}}^3}$

• $3{\mathrm{x}}^2{\mathrm{e}}^{{\mathrm{x}}^3} + 3{\mathrm{x}}^2$

• $3{\mathrm{x}}^2{\mathrm{e}}^{{\mathrm{x}}^3}$

• $3{\mathrm{x}}^3{\mathrm{e}}^{{\mathrm{x}}^3}$

Which of the following functions is a solution of the differential equation ?

${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 - \mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + \mathrm{y} = 0$

• y = 2x - 4

• y = 2x² - 4

• y = 2

• y = 2x

y = ae^mx + be^{- mx} satisfies which of the following differential equations

• $\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{my} = 0$

• $\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{my} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - {\mathrm{m}}^2\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\mathrm{m}}^2\mathrm{y} = 0$

Solution of the differential equation $\frac{\mathrm{dx}}{\mathrm{x}} + \frac{\mathrm{dy}}{\mathrm{y}} = 0$ is

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

• log(x)log(y) = c

• xy = c

• x + y = c

The differential equation representing a family of circles touchmg the Y-axis at the origin is

• ${\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 - 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 + 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 -\hspace{0.17em}{\mathrm{y}}^2 - 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• ${\mathrm{x}}^2 -\hspace{0.17em}{\mathrm{y}}^2 + 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

The general solution of the differential equation (2x - y + 1)dx + (2y - x + 1) dy = 0 is

• x² + y² + xy - x + y = c

• x² + y² - xy + x + y = c

• x² - y² + xy - x + y = c

• x² - y² - xy + x + y = c