Question

If C = 0·003x³ + 0·02x² + 6x + 250 gives the amount of carbon pollution in air in an area on the entry of x number of vehicles, then find the marginal carbon pollution in the air, when 3 vehicles have entered in the area.

Find the general solution of the following differential equation : $y-x{\frac{d y}{d x}}=a\left(y^{2}+{\frac{d y}{d x}}\right).$

Solve the differential equation given as : $\left(\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{d x}{d y}\;=1,x\neq0$

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 equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point $(x_{1},\;y_{1}),$ where x₁ = 2 and y₁ > 0.

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 equations of the normal to the curve y = 4x³ – 3x + 5 which are perpendicular to the line 9x – y + 5 = 0..