Consider the differential equation x^2y\" - 6xy\' + 12y = 0 with general solutio
ID: 3122760 • Letter: C
Question
Consider the differential equation x^2y" - 6xy' + 12y = 0 with general solution y = C_1x^3 + C_2x^4 (a) (i) USE PICARD'S THEOREM to show that the initial value problem {x^2y" - 6xy' + 12y = 0, y(1) = 1, y'(1) = 0, has a unique particular solution on the interval I = (0, infinity) (ii) Determine directly what that particular solution is? (b) (i) SHOW THAT PICARD'S THEOREM fails to show that the initial value problem {x^2y" - 6xy' + 12y = 0, y(0) = 0, y'(0) = 0, has a unique particular solution on the interval I = (-infinity, infinity). (ii) Determine directly if this initial value problem has no particular solution, one particular solution, or infinitely many particular solutions on the interval I = (-infinity, infinity).Explanation / Answer
a) ii) Given : y = C1x3 + C2x4 is the genral solution.
y(1) = C1(1)3 + C2(1)4
y(1) = C1 + C2
But , y(1) = 1
Therefore, C1 + C2 = 1 ........ equation (1)
y' = 3 C1x2 + 4 C2x3
y'(1) = 3 C1(1)2 + 4 C2(1)3
y'(1) = 3 C1 + 4 C2
But , y'(1) = 0
Therefore , 3 C1 + 4 C2 =0 .........equation (2)
From equation (1), we have
C1 = 1 - C2 ......................equation (3)
Substituting the value of C1 in equation (2)
3 ( 1 - C2 ) + 4 C2 =0
3 - 3C2 + 4 C2 =0
3 + C2 =0
C2 = -3
Put this value of C2 in equation 3 , we have
C1 = 1 - C2
C1 = 1 - (-3)
C1 = 1 + 3 = 4
Therefore, y = 4x3 - 3x4 is the particular solution .