Let b, c be constants such that b + c = 1. Determine all the values of r in terms of b, c for which the following differential equation has solutions of the form y = e^rt. (d^2y/dt^2) + b(dy/dt) + cy = 0.

Find the solution of the given initial value problem. y'-2y = e^2t, y(0) = 2.

Find the solution of the given initial value problem. ty'+(t+1)y=t, y(ln2)=1, t>0

Consider the initial value problem y'+2/3y=1-1/2t, y(0)=y0. Find the value of y0 for which the solution touches, but does not cross, the Y-axis.

Show that if a, A are positive constants, and b is any real number, then every solution of the equation y'+ ay = be^-At has the property that y=>0 as t=>infinity.

Let a, b, c, d be constants. Solve the differenatial equation dy/dx = (ay+b)/(cy + d)