Solve $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{y}}^2}{\mathrm{xy} - {\mathrm{x}}^2}$

• y = ce^x/y

• y = ce^-y/x + x

• y = ce^y/x

• xy = ce^y/x

Solve $\mathrm{xcos}\left(\mathrm{x}\right)\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + \mathrm{y}\left(\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right) = 1$

• $\mathrm{y} = \mathrm{xtan}\left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\mathrm{x} = \mathrm{ytan}\left(\mathrm{x}\right) + \mathrm{c}$

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

• $\mathrm{xycos}\left(\mathrm{x}\right) = \mathrm{x} +\hspace{0.17em}\mathrm{c}$

Differential equation of the family of curve y = a cos(µx) + b sin(µx), where a, b are arbitrary constants, is given by

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

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

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

• None of these

The differential equation of all circles which pass through the origin and whose centres lie on y-axis is

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

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

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

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

The differential equation of the family of curve y = Ae^3x + Be^5x, where A, B are arbitrary constants, is

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 8\frac{\mathrm{dy}}{\mathrm{dx}} + 15 = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 8\frac{\mathrm{dy}}{\mathrm{dx}} + 15 = 0$

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

• None of these

306.

Solution of the differential equation $\mathrm{x}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \mathrm{y} + \sqrt{\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)}$ is

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

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

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

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

The differential equation whose solution represents the family y = ae^3x + be^x is given by

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 4\frac{\mathrm{dy}}{\mathrm{dx}} - 3\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 4\frac{\mathrm{dy}}{\mathrm{dx}} - 3\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 4\frac{\mathrm{dy}}{\mathrm{dx}} + 3\mathrm{y} = 0$

• None of the above

Solve $\frac{2\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{y}}{\mathrm{x}} + \frac{\mathrm{y}}{{\mathrm{x}}^2}$

• y = x + $\mathrm{C}\sqrt{\mathrm{xy}}$

• $\mathrm{y} = \mathrm{x} - \mathrm{C}\sqrt{\mathrm{xy}}$

• $\mathrm{y} = \mathrm{x} + \mathrm{Cy}\sqrt{\mathrm{x}}$

• $\mathrm{y} = \mathrm{x} + \mathrm{C}\sqrt{\mathrm{y}}$

Solve $\left(\mathrm{x} +\hspace{0.17em}2{\mathrm{y}}^3\right)\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{y}, \mathrm{y} >\hspace{0.17em}0$

• y = x³ + Cy

• x = y³ + Cy

• y = x³ - Cy

• x = y³ - Cy

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

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

• $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{Cx}$

• $\sin \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = \mathrm{Cy}$

• $\sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) = \mathrm{Cy}$