Question

Consider the following. (A computer algebra system is recommended. Round your answers to four decimal places.) y' = 3 cos t − 6y, y(0) = 0

Consider the following. (A computer algebra system is recommended. Round your answers to four decimal places.) y 3 cos t - 6y, y(0)0 (a) Find approximate values of the solution of the given initial value problem at t 0.1, 0.2, 0.3, and 0.4 using the Euler method with h0.1 y(0.1)0.3 y (0.2)0.4185 y(0.3)0.4614 y(0.4) = 0.4712 (b) Find approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, and 0.4 using the Euler method with h = 0.05 y(0.1) = 0.2548 y(0.2)0.3776 y (0.3)0.4333 y(0.4) = 0.4535 (c) Find approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, and 0.4 using the Euler method with h = 0.025 y(0.1)0.2387 y(0.2)-0.3611 y (0.3) 0.4204 y(0.4)0.4445 (d) Find the solution y- (t) of the given problem y(t)- Evaluate p(t) at t0.1, 0.2, 0.3, and 0.4. (Round your answers to five decimal places.) y(0.1) = 0.23165 y(0.2)= 0.35352 x y(0.3) -0.41425 X y(0.4)= 0.43998 ×

Please solve all parts of d)

the equation and the evaluation of y(0.1)~y(0.4)

Answer 1

We first find solution for y'=3cost-6y.hence we get y(t).then we find y(0.1) by substituting t=0.1.

d) y^1 = 3 cos t_6y, y(0) = 0 y^1 + 6y = 3 cos t integrating factor = e^integral p(t) dt = e^integral 6 dt = e^6t y middot e^6t = integral 3 cos t middot e^6t dt = 3 e^6t (sin(t) + 6 cos t)/37 + c implies y = 3/37 sin t + 18 cos t \37 + c e^-6t pi(t) = 3/37 sin t + 18/37 cos t + c e^-6t y(0) = 0 implies 0 = 18/37 + c c = -18/37 pi(t) = 3/37 sin t + 18/37 cos t + - 18/37 e^-6t y(0.1) = 3/37 sin(0.1) + 18/37 cos(0.1) -18/37 e^-6(0.1) = 0.219647 y(0.2) = 3/37 sin(0.2) + 18/37 cos(0.2) - -18/37 e^-6(0.2) = 0.34024 y(0.3) = 3/37 sin(0.3) + 18/37 cos(0.3) - 18/37 e^-6(0.3) = 0.40649 y(0.4) = 3/37 sin (0.4) + 18/37 cos(0.4) - 18/37 e^-6(0.4) = 0.44932