Use differential equation and given point to find f(x)

Question

Use the differential equation and the given point to find f(x). 

 

dy/dx= (-48)/(3x+5)^2             Point: (-1,3)

Answer 1

Take the integral of dy/dx to get f(x).

 

f(x) = ∫dy/dx

 

f(x) = -48∫(1 / (3x + 5)²)dx

 

 

Let              u = 3x + 5

                 du = 3dx

         (1/3)du = dx

 

Then substituting in,

 

f(x) = (1/3) * -48∫(1 / u²)du

f(x) = -16∫u^-2du

f(x) = -16(-u^-1/1)

f(x) = 16u^-1

 

Substituting back into x,

 

f(x) = 16(3x + 5)^-1 + C

f(x) = [16 / (3x + 5)] + C      ----->  general function

 

 

Then to find C, set f(-1)=3

 

3 = 16 / [3(-1) + 5] + C

3 = (16 / 2) + C

3 = 8 + C

-5 = C

 

Finally, you have

 

f(x) = [16 / (3x + 5)] - 5