Answer to Question #347453 in Algebra for Xhaka

Question

Given that Ae^×+B^-×=4coshx-5sinhx. Determine the values of A and B

Answer 1

Solution

From definitions cosh(x) = (e^x + e^-x)/2, sinh(x) = (e^x – e^-x)/2

So 4cosh(x) – 5sinh(x) = 2(e^x + e^-x) – 2.5(e^x – e^-x) = -0.5 e^x + 4.5 e^-x = -0.5 e^x + e^ln(4.5)-x = -0.5 e^x + e^{-(x- ln(4.5))}

If  Ae^×+B^-× = 4cosh(x) – 5sinh(x) = -0.5 e^x + 4.5 e^-x then A = -0.5,  

B^-× = 4.5 e^-x  => e^-xln(B) = e^-x+ln(4.5)  =>  xln(B) = x – ln(4.5)  =>  ln(B) = 1 – ln(4.5)/x  => B = e^1-ln(4.5)/x  = e / 4.5^1/x  = e 4.5^-1/x  =>  B = B(x) = e 4.5^-1/x    

But it seeems to me that there is mistake in task equality. May be Ae^×+Be^-× = 4cosh(x) – 5sinh(x). In this case A = -0.5, B = 4.5.