Question

(1 point) Find the function satisfying the differential equation 6er and y(0) = 4

Answer 1

Salution The gwen lineal differential equation is y2y=-66L - with y0)=4 The linear differential equation in standard form is given as y' + PCty = 9c+) ☺ where both . pct) and act) are continuous Function of to constant. we solve this equation qualion using integrating factor method. Integrating factor de espald! fo t with pct) = -2 egn and . 90+) = -6et compaire ega we have 801 Integrating factor c 4 = {PCE) at ofagok el se lle 24 multiply egr ④ both sides ky zate y'-24) - et 7-6 ett y ezt ay et seet which can be written as . (y ē25)' = -6 ezt [since, Cy4)' = y (2)'+ y' (€25) = yes ét) + y're?) = -2yéty'e t (yet) = glezt zy z 1

from egn ☺ (yet)' = -6 Afyec). -6 24 - Integrating both sides with respect to t. y e²t) at - f-6 ezt a Sdry ezt) = -6 fet at integration and derivative sign gets cancel to each other. ema serate - ... yệt = -6125)+ c where c is constant of . y 24 = 3 ezt tc - s integration © - also given that Y (o)= 4 ..90)ė = 3 etc yo) = 37C 4 = 37C from egn o 27 3 26 +1 Ya 32RE

Hence yCE)- 37 e E Hence, The function suties fying equation y'a ay =-6eet is l y = 3+ eZE I the differential