Mathematics : Differential Equations

$\mathrm{The} \mathrm{solution} \mathrm{of} \left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)\mathrm{dx} = 2\mathrm{xydy} \mathrm{is} :$

• $\mathrm{c}\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right) = \mathrm{x}$

• $\mathrm{c}\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right) = \mathrm{x}$

• $\mathrm{c}\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right) = \mathrm{y}$

• $\mathrm{c}\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right) = \mathrm{y}$

$\mathrm{The} \mathrm{solution} \mathrm{of} \left(1 + {\mathrm{x}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{xy} - 4{\mathrm{x}}^2 = 0 \mathrm{is} :$

• $3\mathrm{x}\left(1 + {\mathrm{y}}^2\right) = 4{\mathrm{y}}^3 + \mathrm{c}$

• 3y(1 + x²) = 4x³ + c

• 3x(1 + y²) = 4y³ + c

• 3y(1 + y²) = 4x³ + c

$\mathrm{The} \mathrm{solution} \mathrm{of} \frac{\mathrm{dx}}{\mathrm{dy}} + \frac{\mathrm{x}}{\mathrm{y}} = {\mathrm{x}}^2 \mathrm{is} :$

• $\frac{1}{\mathrm{y}} = \mathrm{cx} - \mathrm{xlog}\left(\mathrm{x}\right)$

• $\frac{1}{\mathrm{x}} = \mathrm{cy} - \mathrm{ylog}\left(\mathrm{y}\right)$

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

• $\frac{1}{\mathrm{y}} = \mathrm{cx} - \mathrm{ylog}\left(\mathrm{x}\right)$

The differential equation obtained by eliminating the arbitrary constants a and b from xy = ae^x + be^{- x} is

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

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

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

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

$\mathrm{The} \mathrm{solution} \mathrm{of} \left(\mathrm{x} + \mathrm{y} +\hspace{0.17em}1\right)\frac{\mathrm{dy}}{\mathrm{dx}} = 1 \mathrm{is}$

• y = (x + 2) + ce^x

• y = - (x + 2) + ce^x

• x = - (y + 2) + ce^y

• x = (y + 2)² + ce^y

$\mathrm{The} \mathrm{solution} \mathrm{of} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{y}}^2}{\mathrm{xy} - {\mathrm{x}}^2} \mathrm{is}$

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = \mathrm{kx}$

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = \mathrm{ky}$

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{y}$}\right.} = \mathrm{kx}$

• ${\mathrm{e}}^{\raisebox{1ex}{$ - \mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = \mathrm{ky}$

$\mathrm{The} \mathrm{solution} \mathrm{of} \frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}1 = {\mathrm{e}}^{\mathrm{x} +\hspace{0.17em}\mathrm{y}} \mathrm{is}$

• ${\mathrm{e}}^{ - \left(\mathrm{x} + \mathrm{y}\right)} +\hspace{0.17em}\mathrm{x} + \mathrm{c} = 0$

• ${\mathrm{e}}^{ - \left(\mathrm{x} + \mathrm{y}\right)} -\hspace{0.17em}\mathrm{x} + \mathrm{c} = 0$

• ${\mathrm{e}}^{ \left(\mathrm{x} + \mathrm{y}\right)} +\hspace{0.17em}\mathrm{x} + \mathrm{c} = 0$

• ${\mathrm{e}}^{ \left(\mathrm{x} + \mathrm{y}\right)} -\hspace{0.17em}\mathrm{x} + \mathrm{c} = 0$

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{xy} + \mathrm{y}}{\mathrm{xy} + \mathrm{x}} \mathrm{is}$

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

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

• $\mathrm{x} - \mathrm{y} - \log \left(\frac{\mathrm{cx}}{\mathrm{y}}\right)$

• $\mathrm{y} - \mathrm{x} = \log \left(\frac{\mathrm{cx}}{\mathrm{y}}\right)$

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} - 2\mathrm{y} + 1}{2\mathrm{x} - 4\mathrm{y}} \mathrm{is}$

• (x - 2y)² + 2x = c

• (x - 2y)² + x = c

• (x - 2y)² + 2x² = c

• (x - 2y) + x² = c

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

• $\mathrm{y} = {\mathrm{e}}^{\mathrm{x}}\cos \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{ycos}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}} + \mathrm{c}$

• $\mathrm{y} = {\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{ysin}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}} + \mathrm{c}$