Question

This problem is a very simple example of a differential equation: an equation that relates a function to one or more of its derivatives. You can solve this problem by doing some educated guessing. ("educated" means "remember what we did in the past.") Suppose f is the function that satisfies f(x)-fx) for all T in its domain, and Then f(x) =___________

Answer 1

Answer:

f(x) = 1/x

Step-by-step explanation:

The corrected question is:

Suppose f is the function that satisfies  

f'(x)= - f²(x)  

for all x in its domain, and  

f(1) = 1  

f(x)= ?

Given:

f'(x) = -(f(x))²

This can be written as:

df/dx = -f²

The equation becomes:

-df/f² = dx

Integrating the above equation on both sides we get:

1/f = x + c     where c is constant

So

f(x) = 1/(x+c)

Now given that:

f(1) = 1  

Solving this for c we get:

f(1) = 1 = 1/(1 + c)

1 = 1/ 1+c  

1 = 1 + c

c = 1 - 1

c = 0

Now put this value of c in f(x) = 1/(x+c)

f(x) = 1/(x+0)

f(x) = 1/(x)

f(x) = 1/x

Hence

f(x) = 1/x