Differential Equations · Question 35

Let y = y(x) be the solution of the differential equation : x( e( x))2 dy + ( x - 3y x e( x))dx = 0, x (0, 2 ). If y ( 4 ) = - 1 e 2, then y ( 6 ) is equal to :

1. 2 e(3) - e(4)
2. - 1 e(4)
3. 1 e(4) - e(3)
4. 1 e(3) - e(4)

Answer

(4) 1 e(3) - e(4)

Solution

Let u = e( x) dudx = - x

Dividing the equation by x u2 dx:

dydx + xu2 - 3y xu = 0

dydx - 1u2 dudx + 3yu dudx = 0

dydu + 3uy = 1u2 (linear)

I.F. = e 3udu = u3

y u3 = u du = u22 + c

At x = 4: u = - 12 e 2, y = - 1 e 2

y u3 = ( e 2)28 = u22 c = 0

y = 12u = 12 e( x)

At x = 6: 2 e 32 = e 34 = e(3) - e(4)

y ( 6 ) = 1 e(3) - e(4)

Answer: (4)