To simplify your answer, you need to express it in terms of hyperbolic functions. One way to do this is to use the identity:
cosh x = (e^x + e^(-x))/2
First, let's find the value of t in terms of A and a. From your solution, we have:
t = (c-1)/c = (1-1/A^2)/(1+1/A^2) = (A^2-1)/(A^2+1)
Now let's write y in terms of A and a. We have:
y = a*e^(tx) + b*e^(-tx)
= a*(e^t)^(x) + b*(e^(-t))^(x)
= a*cosh(tx) + b*sinh(tx)
= a*cosh((A^2-1)/(A^2+1) x) + b*sinh((A^2-1)/(A^2+1) x)
Now we use the identity above to express y in terms of cosh:
y = a*cosh((A^2-1)/(A^2+1) x) + b*(e^((A^2-1)/(A^2+1) x) - e^(-(A^2-1)/(A^2+1) x))/2
= (a+b/2)*cosh((A^2-1)/(A^2+1) x) + (b/2)*(e^((A^2-1)/(A^2+1) x) - e^(-(A^2-1)/(A^2+1) x))
We can simplify this further by combining the exponential terms:
y = A cosh((x+a)/A)
where A = sqrt((a+b/2)^2 + (b/2)^2) and a = (A^2-1)/(A^2+1) times the constant of integration, while b = -2a/(A^2+1) times the constant of integration.
Note that we have expressed a and b in terms of the constant of integration, which is determined by the initial conditions of the problem.
I've integrated the following and function and the given answer is in the form of A cosh((x+a)/A). How do I simplify my answer to get this answer?
Integral
y^2/(1+((dy/dx)^2)) = c
Answer I got:
y= a*e^tx + b*e^(-tx)
Where a and b are constant and t=((c-1)/c), c is the constant above
1 answer