31.

Find the general solution of (x + log(y))dy + ydx = 0

The general solution of the differential equation

$\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} +\hspace{0.17em}8\frac{d\mathrm{y}}{d\mathrm{x}} + 16\mathrm{y} = 0$ is

• (A + B)e^5x

• (A + Bx)e^{- 4x}

• (A + Bx²)e^4x

• (A + Bx⁴)e^4x

If x² + y² = 4, then $\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{x}$ is equal to

• 4

• 0

• 1

• - 1

$\int \frac{{\mathrm{x}}^3\mathrm{dx}}{1 + {\mathrm{x}}^8}$ is equal to

• $4{\tan }^{-1}\left({\mathrm{x}}^4\right) + \mathrm{C}$

• $\frac{1}{4}{\tan }^{-1}\left({\mathrm{x}}^3\right) + \mathrm{C}$

• $\mathrm{x} +\hspace{0.17em}4{\tan }^{-1}\left({\mathrm{x}}^4\right) + \mathrm{C}$

• ${\mathrm{x}}^2 +\hspace{0.17em}\frac{1}{4}{\tan }^{-1}\left({\mathrm{x}}^4\right) + \mathrm{C}$

The degree and order of the differential equation

y = $\mathrm{x}{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 + {\left(\frac{d\mathrm{x}}{d\mathrm{y}}\right)}^2$ are respectively

• 1, 1

• 2, 1

• 4, 1

• 1, 4

The general solution of the differential equation ${\log }_{\mathrm{e}}\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right) = \mathrm{x} + \mathrm{y}$ is

• e^x + e^{- y} = C

• e^x + e^y = C

• e^y + e^{- x} = C

• e^{- x} + e^{- y} = C

If $\mathrm{y} = \frac{\mathrm{A}}{\mathrm{x}} + {\mathrm{Bx}}^2,$ then ${\mathrm{x}}^2\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2}$ is equal to

• 2y

• y²

• y³

• y⁴

The solution of $\frac{d\mathrm{y}}{d\mathrm{x}} = \frac{\mathrm{y}}{\mathrm{x}} + \tan \left(\frac{{\displaystyle \mathrm{y}}}{{\displaystyle \mathrm{x}}}\right)$ is

• x = c$\sin \left(\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right)$

• x = c$\sin \left(\mathrm{xy}\right)$

• y = c$\sin \left(\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.\right)$

• xy = c$\sin \left(\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{y}$}\right.\right)$

Integrating Factor (IF) of the differential equation $\frac{d\mathrm{y}}{d\mathrm{x}} - \frac{3{\mathrm{x}}^2\mathrm{y}}{1 + {\mathrm{x}}^3} = \frac{{\displaystyle {\sin }^2\left(\mathrm{x}\right)}}{1 + \mathrm{x}}$

• ${\mathrm{e}}^{1 + {\mathrm{x}}^3}$

• $\log \left(1 + {\mathrm{x}}^3\right)$

• 1 + x³

• $\frac{1}{1 + {\mathrm{x}}^3}$

The differential equation of y = ae^bx (a and b are parameters) is

• yy₁ = y₂²

• yy₂ = y₁²

• yy₁² = y₂

• yy₂² = y₁