Question

Find the equations of tangents to the curve 3x² – y² = 8, which passes through the point $\left({\frac{4}{3}},\,0\right).$

Solve the following differential equation. ${\sqrt{1+x^{2}+y^{2}+x^{2}y^{2}}}+x y{\frac{d y}{d x}}\,=0$

Solve the differential equation : ${\frac{d y}{d x}}-3y\cot x=\sin2x\operatorname{given}y=2,{\mathrm{~when~}}x={\frac{\pi}{2}}.$

Solve the differential equation $x\;\frac{d y}{d x}\;+y=$ x cos x + sin x, given that y = 1 when $x={\frac{\pi}{2}}$

Obtain the differential equation of all the circles of radius r.

Show that the normal at any point θ to the curve x = a cos θ + aθ sin θ, y = a sin θ – aθ cos θ is at a constant distance from the origin.

Find the particular solution of the differential equation $\left(1+x^{2}\right){\frac{d y}{d x}}=\left(e^{{\mathrm{mtan}^{\perp}x}}-y\right)$ given that y = 1 when x = 0.