You must use Python to solve all of the following problems, except the first one, for which you may use dfield. You will need to run the codes and print both the codes and the output (solution). 1) Draw a direction field for the equation y'=y+x-y^2 for -5 < x, y <5, and include the integral curves passing through (x_0,y_0)=(0,0), (2,3), and (-2,-3).
2) Solve the initial value problem (x^2-1)y' + 2xy^2 = 0, y(sqrt(2)) = 1. Give an explicit solution.
3) Solve the ODE xy' + 2y = sin(x), x > 0. Give an explicit solution.
4) Solve the IVP (1/x + 2y^2x)dx + (2yx^2 - cos(y))dy = 0, y(1) = pi. Give an implicit solution.
5) A tank contains 25 lb of salt dissolved in 50 gal of water. Brine containing 4 lb/gal enters the tank at a rate of 2 gal/min and leaves the tank at the same rate.
6) An object cools in 10 min from 100C to 60C. The surroundings are at a temperature of 20C. When will the object have cooled to 25C?7) Solve the differential equation y'' + 4y = sin(2t) - cos(2t).
(a) Determine the amount y of salt in the tank at time t
(b) When is there 150 lb of salt in the tank?