solve for differential equation all possible solutions.

Question

dy/dt+t=2y

Answer 1

This is a standard linear first order differential equation in the form y' - 2y = -t

You can use the integration factor e^-2t to multiply through and obtain (using product rule in reverse):

d/dt (e^-2ty) = te^-2t which can be written as the integral, solving for y:

y - e^2t(int(te^-2tdt) + C) This equation is in every D Eqn text as a general equation with f(f) for t and the negative integral of the coefficient function (here -2) from in front of the y term)

integrating by parts and multiplying out:

y = (1/2)t + 1/4 + Ce^2t

There are other ways to do this, but this is the explicit answer.

You can also find the homogeneous y_h (equate right-hand side to 0 ) and then use the method of undetermined coefficients to determine the particular solution (y_p). The solution y = y_h + y_p