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
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.
To simplify your answer to the form A cosh((x+a)/A), you can follow these steps:
1. Start with the answer you obtained: y = a*e^tx + b*e^(-tx).
2. Rewrite the answer using hyperbolic functions: y = (a/2)*(e^(tx) + e^(-tx)) + (b/2)*(e^(tx) - e^(-tx)).
3. Notice that e^(tx) + e^(-tx) is the definition of the hyperbolic cosine function: cosh(tx).
4. Also, e^(tx) - e^(-tx) can be rewritten as sinh(tx).
5. Substitute these values back into the answer: y = (a/2)*cosh(tx) + (b/2)*sinh(tx).
6. To match the form A cosh((x+a)/A), you can let A = c and (x+a)/A = tx.
7. Solve for a and x to find the values of a and x, then substitute them into the equation.
Therefore, the simplified form of your answer is:
y = (a/2)*cosh((x+a)/A) + (b/2)*sinh((x+a)/A),
where a and b are constants, A = c, and c is the constant from the original integral equation.