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Find y in terms of x for the equation 2x(dy/dx) + 4y = 8x^2

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( integral(2/x) dx ) = exp( 2 ln(x) ) = x 2 therefore d( (x ...
TE
7730 Views

How can I find the explicit formula for the inverse of sinh?

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. T...
SG
3525 Views

using an integrating factor, find the general solution of the differential equation dy/dx +y(tanx)=tan^3(x)sec(x)

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 multiplied ...
TR
6138 Views

Solve (z-i)+(z+i)+(z-1)+(z-1)

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 ta...
YZ
4565 Views

For f(x) = (3x+4)^(-2), find f'(x) and f''(x) and hence write down the Maclaurin series up to and including the term in x^2.

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)+f'(0)x+1/2 f''(0)x^2;...
JM
4336 Views