Differential Equations

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = {\mathrm{e}}^{\mathrm{y}}\left({\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \mathrm{x}} + 2\mathrm{x}\right)$ is

• ${\mathrm{e}}^{- \mathrm{y}} = {\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \mathrm{x}} + {\mathrm{x}}^2 + \mathrm{C}$

• ${\mathrm{e}}^{- \mathrm{y}} = {\mathrm{e}}^{- \mathrm{x}} - {\mathrm{e}}^{\mathrm{x}} - {\mathrm{x}}^2 + \mathrm{C}$

• ${\mathrm{e}}^{- \mathrm{y}} = - {\mathrm{e}}^{- \mathrm{x}} - {\mathrm{e}}^{\mathrm{x}} - {\mathrm{x}}^2 + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{y}} = {\mathrm{e}}^{- \mathrm{x}} + {\mathrm{e}}^{\mathrm{x}} + {\mathrm{x}}^2 + \mathrm{C}$

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

• 3 and 1

• 3 and 3

• 1 and 3

• 3 and 2

The differential equation representing the family of curves given by y = ae^- 3x + b, where a and b are arbitrary constants, is

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 3\frac{\mathrm{dy}}{\mathrm{dx}} - 2\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 3\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 3\frac{\mathrm{dy}}{\mathrm{dx}} - 2\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 3\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

An integrating factor of the differential equation xdy - ydx + x²e^xdx = 0 is

• $\frac{1}{\mathrm{x}}$

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

• $\sqrt{1 + {\mathrm{x}}^2}$

• x

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

• y = log(x) + C

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

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

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

The solution of the differential equation $\log \left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{y}}{\mathrm{x}} = \sin \left(2\mathrm{x}\right)$ is

• $\mathrm{ylog}\left|\mathrm{x}\right| = \mathrm{C} - \frac{1}{2}\cos \left(\mathrm{x}\right)$

• $\mathrm{ylog}\left|\mathrm{x}\right| = \mathrm{C} + \frac{1}{2}\cos \left(2\mathrm{x}\right)$

• $\mathrm{ylog}\left|\mathrm{x}\right| = \mathrm{C} - \frac{1}{2}\cos \left(2\mathrm{x}\right)$

• $\mathrm{xylog}\left|\mathrm{x}\right| = \mathrm{C} - \frac{1}{2}\cos \left(2\mathrm{x}\right)$

If xy = A sin(x) + B cos(x) is the solution of  the differential equation $\mathrm{x}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 5\mathrm{a}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = 0$, then the value of a is

• $\frac{2}{5}$

• $\frac{5}{2}$

• $\frac{- 2}{5}$

• $\frac{- 5}{2}$

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3{\mathrm{e}}^{2\mathrm{x}} + 3{\mathrm{e}}^{4\mathrm{x}}}{{\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \mathrm{x}}}$ is

• $\mathrm{y} = {\mathrm{e}}^{3\mathrm{x}} + \mathrm{C}$

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

• $\mathrm{y} = {\mathrm{e}}^{\mathrm{x}} + \mathrm{C}$

• $\mathrm{y} = {\mathrm{e}}^{4\mathrm{x}} + \mathrm{C}$

The order and degree of the differential equation of the family of circles of fixed  radius r with centres on the y-axis, are respectively

• 2, 2

• 2, 3

• 1, 1

• 1, 2

The solution of the differential equation (kx - y² ) dy = (x² - ky) dx is

• x³ - y³ = 3kxy + C

• x³ + y³ = 3kxy + C

• x² - y² = 2kxy + C

• x² + y² = 2kxy + C