Find the general solution to the differential equation.

Question

a. dy/dt = 2y-t

b. dy/dt=3y+2t

 

Please explain and show all working. Thank you ;)

Answer 1

a. dy/dt - 2y = -t

 

    This is a first order linear differential equation

 

    Multiply the equation by the integrating factor e^∫(-2)dt = e^-2t

 

    e^-2t(dy/dt) - 2e^-2ty = -te^-2t

 

    By the product rule, the left side of the equation is (e^-2ty)'

 

    So, (e^-2ty)' = -te^-2t

 

    Integrate both sides to get e^-2ty = -∫te^-2tdt

 

                                                        u = t      dv = e^-2tdt

                                                       du = dt     v = (-1/2)e^-2t

 

    e^-2ty = (1/2)te^-2t - (1/2)∫e^-2tdt

 

    e^-2ty = (1/2)te^-2t + (1/4)e^-2t + C

 

         y = (1/2)t + 1/4 + Ce^2t

 

    ------------------------------------------------------------------------------

    b.  dy/dt - 3y = 2t

 

         The integrating factor is e^-3t.  Proceed as in part a.