Question

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

Let R be a relation defined on the set of natural numbers N as follow : R = {(x, y) : $x\in N,y\in N$ and 2x + y = 24} Find the domain and range of the relation R. Also,find if R is an equivalence relation or not.

If the radius of sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.

Find the particular solution of the differential equation ${\frac{d y}{d x}}\ =1+x+y+x y$ given that y = 0 when x = 1.

The total expenditure (in ₹) required for providing the cheap edition of a book for poor and deserving students is given by R(x) = 3x² + 36x, where x is the number of set of books. If the marginal expenditure is defined as ${\frac{d R}{d x}},$ write the marginal expenditure required for 1200 such sets.

Evaluate : ($\large\textstyle\int(3x-2){\sqrt{x^{2}+x+1}}\,d x~.$

Prove that the function f : [0, ∞) → R given by f(x) = 9x² + 6x – 5 is not invertible. Modify the Codomain of the function f to make it invertible, and hence find f⁻¹.