91Ó°ÊÓ

Explain why the functions with the given graphs cannot be solutions of the differential equation \(\frac{d y}{d t}=e^{t}(y-1)^{2}\)

In a tank are 100 litres of brine containing \(50 \mathrm{~kg}\) of dissolved salt. Water runs into the tank at the rate of 3 litres per minute, and the concentration is kept uniform by stirring. How much salt is in the tank at the end of one hour if the mixture runs out at a rate of 2 litres per minute?

Show that the tangents to all integral curves of the differential equation \(y^{\prime}+y \tan x=x \tan x+1\) at the points of intersection with the y-axis are parallel. Determine the angle at which the integral curves cut the \(\mathrm{y}\)-axis.

A curve is such that the intercept a tangent cuts off on the ordinate axis is half the sum of the coordinates of the tangency point . Form the differential equation and obtain the equation of the curve if it passes through \((1,2)\).