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

Form the differential equation of all circles which is tough the x-axis at the origin.

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

If E and F are independent events, then show that (i) E and F' are independent events. (ii) E' and F are also independent events.

If P(E) = 1/2, and P(F) = 1/5 find P(E U F)' if E and F are independent events.

If P(F) = 0·35 and P(E U F) = 0·85 and E and F are independent events. Find P(E).

Obtain the differential equation of the family of circles passing through the points (a, 0) and (– a, 0).

Show that the solution of differential equation : $y=2(x^{2}-1)+c e^{-x^{2}}\,\mathrm{is}\,\frac{d y}{d x}+2x y-4x^{3}=0$

The radius r of a right circular cylinder is decreasing at the rate of 3 cm/min. and its height h is increasing at the rate of 2 cm/min. When r =7 cm and h = 2 cm, find the rate of change of the volume of cylinder. $[\mathrm{Use}\:\pi=\:\frac{22}{7}\:]$

The radius r of a right circular cylinder is increasing uniformly at the rate of 0.3 cm/s and its height h is decreasing at the rate of 0.4 cm/s. When r = 3.5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. $[\mathrm{Use}\:\pi=\:\frac{22}{7}\:]$

The radius r of the base of a right circular cone is decreasing at the rate of 2 cm/min. and height h is increasing at the rate of 3 cm/min. When r = 3.5 cm and h = 6 cm, find the rate of change of the volume of the cone. $[\mathrm{Use}\:\pi=\:\frac{22}{7}\:]$

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.

Three cards are drawn without replacement from a pack of 52 cards. Find the probability that (i) the cards drawn are king, queen and jack respectively. (ii) the cards drawn are king, queen and jack.

Verify that ax² + by² = 1 is a solution of differential equation $x(y y_{2}+y_{1}^{2})=y y_{1}.$

A balloon, which always remains spherical, has a variable diameter ${\frac{2}{3}}(3x+1)$ . Find the rate of change of its volume with respect to x.