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The exponential function is defined as a power series, which we may (for reasons that are beyond the scope of A-level) differentiate term by term to get another power series. The general term of the serie...

this is an example of an integration by parts problem, we must use integration by parts to evaluate this integral;although this would not be entirely obvious as the integral does not seem to be the produc...

(2x)/2= 2x^-2d2y/dx2= -4x^-1 - 1/4

So we have to integrate these 3 terms one by one. To integrate 2x⁵ you add 1 to the power of x, so that becomes 6, you then divide the coefficient by the new power of x. So 3/6=1/2 giving the i...

differentiate both side with respect to x : (dy/dx)e^(-2x)+y(-2e^(-2x)) = 2+2y(dy/dx)
rearrange it : (-2y+e^(-2x))(dy/dy) = 2 + 2ye^(-2x) ==> dy/dx = ( 2 + 2ye^(-2x) ) / ( -2y+e^(-2x) )