Solution to differential equation?

Question

Can someone help me with this?

I must find the solution to the differential equation:

dy/dx = sin(pie x) given the initial condition that y(2)=2.

Answer 1

dy/dx = sin(πx)
Integrating both sides with respect to x
∫dy/dx dx = ∫sin(πx) dx
∫dy = ∫sin(πx) dx
y = -cos(πx) / π + c
Now y(2) = 2 so
2 = -cos(2π) / π + c
2 = -1/π + c
c = 2 + 1/π
So
y = -cos(πx) / π + 2 + 1/π

Answer 2

dy/dx = sin(πx)
=> dy = sin(πx) dx
Integrating
y = - (1/π) cos(πx) + c
2 = - (1/π) cos(2π) + c
=> c = 2 + 1/π
=> y = - (1/π) cos(πx) + 2 + 1/π

Answer 3

y = ∫ sin (πx) dx
y = (- 1/π) cos(πx) + C
2 = (- 1/π) cos 2π + C
2 = (-1/π) + C
C = 2 + 1/π
y = (-1/π) cos πx + (2 + 1/π)