Question

Please help me with this Differential equation question

1 point) Gwennenst onder MP y, +Jy-(3,3, 021<i with init al coration yo) . 4. 3, x2l (1) Find the explicit solution on the interval 0 x < 1 y(x) = (2) Find the lim r) (3) Then find the explicit solution on the interval x 21 y(x)-

Answer 1

Given First order I v p's y^' + 3y = {3, 0 lessthanorequalto x < 1 -3, x greaterthanorequalto 1 , y(0) = 4 -(1) (i) Now we find explicit solution on the interval 0 lessthanorequalto x < 1 from (1), y^' + 3y = 3, y(0) = 4 implies dy/dx = 3 - 3y implies dy/dx = -3(y - 1) , variable separable implies integral dy/dx = integral -3 dx implies log(y - 1) = -3x + c -(2) sin y(0) = 4 implies log(4 - 1) = -3(0) + c implies c = log 3 sub in (2) implies log(y - 1) = -3x + log 3 implies log(y - 1/3) = -3x implies y - 1/3 = e^-3x implies y - 1 = 3 e^-3x implies y = 1 + 3 e^-3x (ii) x rightarrow 1^limit y(x) = x rightarrow 1^limit 1 + 3 e^-3x = 1 + 3 e^-3x = 1 + 3 e^-3 = 1.493