Question

a) Solve the IVP using either variation of parameters or integration factor. Clearly indicate what the varying parameter is if you use variation of parameters or what the integration factor is if you use that method. Also, indicate the general solution to the homogeneous equation. dy 1 = sin(t) – y, yco) = dt b) Draw the direction field and draw in the graph of the particular solution that you found.

Answer 1

Advanced math:- والے de ولے TA (a) Siven dfferential equation Sinct) - 9 yo) / Integrating Facter rethod - = Sinct)- y at dety - Sc) -> Comparing the above equation with dy + PC) y = QCE) det PC) :: CH): Sinct) Integrating Factor (LF) - Spot lidt T-F by et In both sides. equation Multiplied +y. et = = Sinct) et We now (yet) = ye+ et des produer pule et sinct) dt dcyet) Sinct) dt -

Apply integration on both fides of equation Jolyet) = J et stact) at yet I sint - cast] to seat sin (bttc) dt = a sin bt +c) - bcosto et [Sint - Cort] + 20 Sint-cost +acet yet) This is the general Solution Given gcoi = 1 / 2 ycol = 2 → Stoco) - GB(0) + 20€° = 0-1 +20 20=1+ Therefore, yet): Sint-Cast + no (1) sorry, I have knowledge about drawing direction field of differential equation. I want to skip it because I don't want to give you uyong answer