Solve for x y=(e^x-e^(-x))/2

Problem

y=(ex−e(−x))/2

Solution

1. Multiply both sides by 2 to clear the fraction.

2*y=ex−e(−x)

2. Substitute u=ex to transform the equation into a quadratic form, noting that e(−x)=1/u

2*y=u−1/u

3. Multiply the entire equation by u to eliminate the denominator.

2*y*u=u2−1

4. Rearrange the terms into a standard quadratic equation in terms of u

u2−2*y*u−1=0

5. Apply the quadratic formula to solve for u where a=1 b=−2*y and c=−1

u=(−(−2*y)±√(,(−2*y)2−4*(1)*(−1)))/(2*(1))

6. Simplify the expression under the radical and divide by 2

u=(2*y±√(,4*y2+4))/2

u=(2*y±2√(,y2+1))/2

u=y±√(,y2+1)

7. Discard the negative root because u=ex must always be positive, and y−√(,y2+1) is always negative.

ex=y+√(,y2+1)

8. Take the natural logarithm of both sides to isolate x

x=ln(y+√(,y2+1))

Final Answer

x=ln(y+√(,y2+1))

Want more problems? Check here!