If dy/dx=2xy and y(0)=1.
what is y(1) equal to?

dy/dx = 2xy 
dy/y = (2x)dx 
INT(1/y)dy = INT(2x)dx 
ln y + c = x^2 + d 
ln y = x^2 + constant 
y = e^(x^2 + constant) 

1 = e^constant ... so constant=0 ... so y = e^(x^2) 

Verification: 

y = e^(x^2) 
dy/dx = 2x*e^(x^2) = 2xy 
and e^(0^2) = e^0 = 1

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