4. This question is about the variation of parameters we talked in class. The following parts will help you understand the nature/proof of this method and lead you to the famous so-called Green's...

Extracted text: 4. This question is about the variation of parameters we talked in class. The following parts will help you understand the nature/proof of this method and lead you to the famous so-called Green's function, from which you are just one step away. (a) First find the general solution of y" – 5y' + 6y = 2e* by using the Undetermined Coefficient Method (namely, using the table we build in class by observing the type of the right-hand side function). (b) tion of Parameter Method. Use the information you already obtained in the previous step, solve (3) by variation of parameters. However, do not use the formula directly. Read and mimic steps from eq. (6) to (10) in your textbook on page 161, repeat the process by yourself. Namely, rephrase and apply eq.(6) to (10) to your considered question. Now we want to compare Undetermined Coefficient Method with Varia- y" + P(x)y + Q(x)y= f(x) (6) Yp (x) = u1 (x)y1 (æ)+u2(x)y2(x) (7) =nu + y2u½] + P[y1 u{ + Y2u½] + ¥{u + ½u½ = f(x). (8) W1 Y2 f (x) W2 and u, Y1 f (æ) W W W W Y1 W = Y2 Y2 Y1 W1 = (10) W2 | y1 f(x)


Extracted text: -dt + y2(x) / ² (c) Green's function by yourself. You are now in front of the door and one step close to derive the famous Read eq.(7) to (11) on page 173, 174 and Theorem 4.8.1 on page 175. Then use Green's function, i.e. eq.(10) to solve IVP y" – 5y' + 6y = 2e* y(0) = 0, y'(0) = 0 Yp (x) = u1 (x)y1 (x) + u2 (x)y2 (x). (7) Y2 (w)f(x) Y1 (x)f(x) u (x) = - u (x) (8) = W W / -Y2 (t) f(t) W(t) Y1 (t) f(t) -dt Yp (x) = Y1 (x) dt + y2(x) W (t) (9) yı (t)y2 (x) f(t) dt, -Y1 (x)y2 (t) W (t) Y1 -f(t) dt + W (t) Yp (a) = | G(x, t)f(t) dt. (10) Y1 (t)y2 (x) – Y1 (x)y2 (t) W (t) G(x, t) : (11)