Question

If the basis of a third order De are: y = 1,7=e- *.y=e6x. Then the corresponding De is: Select one: a. xy" - 4x?y"-5xy'+y=0 b. 5y"-y"+y=0 C. Yu-y"+5y=0 d. y"-4y"-5y=0

Answer 1

Given that the basis anders of third DE que : ) Y = 1 → Y2 = x 1 Y3 = 25% We have we andea DE :) to find the correspo esponding DE . know that if Yo Y and Y₃ are the basis of third then Y = 417,+ 6y,+973 is the general solution of the correspo Differential equation [ where a, 3 are sponding arbinary constants For the given problem, Y=1 onder differential equation . So, y = G Y, +(272 +13 43 is the solution of the differential equation 4, 42 and 43 tary constants. Y₂ = ex and /= e 5x are the basis of a third corresponding eu here ar as bila Thus, y = 61.1 +420-?+ ce (з е 5x [Rulting the vawes of yourz] and Y₃ y = (y +626-97 +3e5x Differentiating with respect to a have y = f (atze-*+ Gesx) [:*() = y']

or y'= (0-60** + 56ge5e) [:$(20x) = 2847] ly'= -6e-² 563 e5x -C have Differentiating y' with respect to x (-624-+563 25x) Ce-> +2563e5x [ se (Ve) =y"] Y" Again, differentiating y" with respect to x, we have, y" ad (624+*+253254) 38 ( Y")= v"] Y" -2,2-2 + 125C325x Now, we لامع Se that, the values] y!" – 4y" - 5y' =6407_125055* )-4 (64+256325x) 5 (-24*+55254) (purting =-628-2 +125(3 452-4422-7-100 25x+562677 - 25€325x +(4x2_4647+566->) + (12563657-160 (9252_256325) • (-556**+564-2) + (125625* - 13503254) 0 +0 Thus , we get y!!! - 4y" - 5y' = 0 -W) the third onders So, y = 0, +628-²+ez esx differential equation satisfies y!! - 4y" - 5y'=0

y = 4, +(22-7 + (3 e 5% is solution of the DE (2) basis whose are Vi=) Hence, the corresponding DE Y2 = e-> and is Y3 = e52 -(d} yll – 4 y!! - 5y' = 0 .