Question

Find the sum of the order and the degree of the following differential equations : ${\frac{d^{2}y}{d x^{2}}}+{3{\sqrt\frac{{d y}}{d x}}}+(1+x)=0$

The position vectors of points A, B and C are $λ\stackrel{\wedge}{i}+3\stackrel{\wedge}{j}$,$12\stackrel{\wedge}{i}+μ\stackrel{\wedge}{j}$ and $11\stackrel{\wedge}{i}-3\stackrel{\wedge}{j}$ respectively. If C divides the line segment joining A and B in the ratio 3 : 1, find the values of λ and μ.

The total revenue received from the sale of x units of a product is given by R(x) = 3x² + 36x + 5 in rupees. Find the marginal revenue when x = 5, where by marginal revenue we mean the rate of change of total revenue with respect to the number of items sold at an instant.

From the differential equation of equation y = a cos 2x + b sin 2x, where a and b are constant.

Find : $\textstyle\int$$\frac{1}{\sqrt{x^{2}-4\,x}}\,d x$

Find : $\textstyle\int$${\sqrt{x^{2}-2x}}\ d x$

The volume of a sphere is increasing at the rate of 8 cm³/sec. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.