Prove that x² – y² = c(x² + y²)² is the general solution of the differential equation (x³ – 3xy²)dx = (y³ – 3x²y) dy, where C is a parameter.

322.

Find the differential equation representing the family of curves y = ae^{bx + 5} , where a and b are arbitrary constants.

If y = sin (sin x), prove that $\frac{{\mathrm{d}}^2 \mathrm{y}}{{\mathrm{dx}}^2} + \tan \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} {\cos }^2 \mathrm{x} = 0$

Find the particular solution of the differential equation e^x tan ydx + (2 -e^x) sec² ydy = 0, given that $y = \frac{\pi }{4}$ when x = 0

Find the particular solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + 2 \mathrm{y} \tan \mathrm{x} = \sin \mathrm{x}$, given that y = 0 when $\mathrm{x} = \frac{\mathrm{\pi }}{3}$

If $({\mathrm{x}}^2 + {\mathrm{y}}^2{)}^2 = \mathrm{xy}, \mathrm{find} \frac{\mathrm{dy}}{\mathrm{dx}}$

If $\mathrm{x} =\mathrm{a} (2\mathrm{\theta } - \sin 2\mathrm{\theta }) \mathrm{and} \mathrm{y} = \mathrm{a}(1- \cos 2\mathrm{\theta }), \mathrm{find}\frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{when} \mathrm{\theta } =\frac{\mathrm{\pi }}{3}$

Differentiate the following with respect of x:

$\mathrm{y} = {\tan }^{-1} \left(\frac{\sqrt{1 + \mathrm{x}} - \sqrt{1 - \mathrm{x}}}{\sqrt{1 + \mathrm{x}} + \sqrt{1 - \mathrm{x}}} \right)$

Solve the following differential equation:
(x² − y²⁾ dx + 2xy dy = 0   given that y = 1 when x = 1

Solve the following differential equation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x} ( 2\mathrm{y} - \mathrm{x} )}{\mathrm{x} ( 2\mathrm{y} + \mathrm{x})}$,   if y = 1 when x = 1