Question

Find the equation of the normal at the point (am², am³) for the curve ay² = x³.

Find the particular solution of the following differential equation : $x y{\frac{d y}{d x}}\,=(x+2)\,(y+2);y=-1,{\mathrm{when}}\,x=1.$

Find the particular solution of the differential equation $(1+e^{2x})d y+(1+y^{2})e^{x}d x=0,$ given that y = 1, when x = 0.

Show that the relation R on the set Z of all integers defined by (x, y) ∈ $R\Leftrightarrow(x-y)$ is divisible by 3 is an equivalence relation.

Solve the differential equation : $(1+x^{2}){\frac{d y}{d x}}\ +y=e^{\tan^{-1}x}.$

Find the general solution of the differential equation : $(x-y){\frac{d y}{d x}}\,=x+2y.$

Find the particular solution of the differential equation ${\frac{d y}{d x}}={\frac{x\left(2\log x+1\right)}{\sin y+y\cos y}}$ given that $y={\frac{\pi}{2}}$ where x = 1.