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divide through by 2x to get: dy/dx + 2y/x = 4x         this is now in the form of dy/dx + P(x)y = Q(x)

intergrating factor = exp( integral(P(x)) dx ) = exp( integr...

Write y = sinh^(-1)(x) ie x = sinhy. Then writing this in terms of exponentials and multiplying by 2 we will get 2x = exp(y) - exp(-y). Multiply by exp(y) and rearrange to obtain exp(2y) - 2xexp(y) -1 = 0...

This is a first order differential equation, but initially it is not solvable as both the X&Y's can't be separated from eachother directly. An integrating factor is a specific value that when multipli...

Since we are dealing with complex numbers and taking its modulus, we can rewrite (z-i)=((-1)(i-z))=(i-z) doing the same for (z-1)=(1-z) we get (i-z)+(z+i)+(1-z)+(z-1)=(i+i+z-z+1+1+z-z) =(2i+2)=4 as we are...

f'(x)=-2(3x+4)^(-3) * 3 = -6(3x+4)^(-3);
f''(x)= 18(3x+4)^(-4) * 3 = 54(3x+4)^(-4);
both found by using the chain rule for differentiation.

Then Maclaurin series up to x^2 is: f(x)=f(0)...