Question: Find y as a function of x given that d^2y/dx^2 = 6x-4 (y= 4 when x= 1) and (y= 2 when x= -1)
The DE is,
d^2y/dx^2 = 6x - 4 -- integrate both sides wrt x.
dy/dx = 3x^2 - 4x + C1 -- integrate both sides wrt x again.
y(x) = x^3 - 2x^2 + C1.x + C2
Now use the initial conditions
y(1) = 4
4 = 1^3 - 2*1^2 + C1*1 + C2
4 = 1 - 2 + C1 + C2
C1 + C2 = 5 ------------------- (1)
y(-1) = 2
2 = (-1)^3 - 2(-1)^2 + C1*(-1) + C2
2 = -1 - 2 - C! + C2
C1 - C2 = -5 -----------------(2)
Adding together eqns (1) and (2),
2C1 = 0
C1 = 0, C2 = 5
Finally, y(x) = x^3 - 2x^2 + 5