Question

Solve the following differential equation : $(x^{2}-1){\frac{d y}{d x}}~+2x y={\frac{2}{x^{2}-1}},~~x\neq{\bf1}$

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

Find the general solution of the differential equation: ${\frac{d x}{d y}}\;=\;{\frac{y\tan y-x\tan y-x y}{y\tan y}}$

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

Using properties of determinants, prove the following : $\begin{vmatrix}a & a² & bc \\ b & b² & ca \\ c & c² & ab\end{vmatrix}$ = $(a-b)(b-c)(c-a)(b c+c a+a b).$

Find the particular solution of the differential equation x (1 + y²) dx – y (1 + x²) dy = 0, given that y = 1, when x =0.

Find the equations of the normal to the curve y = x³+ 2x + 6, which are parallel to line x + 14y + 4 = 0.