Question

Find f by solving the initial-value problem. 6/sqrtx, f(9)=39

Answer 1

Applying the initial condition to determine the constant, will get f(x) = 12sqrt(x) + 3.

Step-by-step explanation:

To find the function f by solving the initial-value problem 6/sqrt(x), with the initial condition f(9)=39, we need to perform integration.

We integrate the given function 6/sqrt(x) with respect to x and then apply the initial condition to solve for the constant of integration.

The integration of 6/sqrt(x) with respect to x is:

• Let u = sqrt(x) which means that x = u^2 and dx = 2u du.
• Substitute x and dx in the integral to get 6/u multiplied by 2u du.
• After simplification, the integral becomes 12 du.
• Now, integrate 12 with respect to u to get 12u + C.
• Substituting back for u, we get f(x) = 12sqrt(x) + C.
• Apply the initial condition f(9)=39 to find the value of C.
• Since f(9) = 12sqrt(9) + C = 36 + C, then C = 39 - 36, which gives C = 3.
• The function f is f(x) = 12sqrt(x) + 3.