The tank in form of the right circular cylinder of radius 2 ft and height 10ft is standing on one of its bases. Tank is originally full of water and water leaks from the circular hole of radius 1/2 inch at its bottom. Determine relationship between rates volume and height change with respect to time t. Then find differential equation for height of water at time t. Ignore friction and contraction of water at hole and g=32ft/s^2.
i) For autonomous first-order differential equation, y'=y^2+3y-18 determine critical points and phase portrait. Categorize stability of each critical point.
ii) Determine solution to initial value problem, 2xy^2+(x^2-1)y'=0; y(0)=1.
iii) Determine integrating factor for differential equation and solve it, sec(x)y' +y-1=0.