Express y as a function of x. C is a positive number.
3 ln y= 1/2ln (2x+1) - 1/3ln (x+4) + ln C
ln y = (1/6)ln (2x+1) -(1/9)ln (x+4) + (1/3) ln C
y = e^ln y = e^[(1/6)ln (2x+1) -(1/9)ln (x+4) + (1/3) ln C]
= C^(1/3)* (2x+1)^(1/6) /(x+4)^(1/9)
To express y as a function of x, we need to eliminate the logarithms in the equation.
First, we can use the properties of logarithms to simplify the equation:
ln y = (1/6)ln (2x+1) -(1/9)ln (x+4) + (1/3) ln C
Using the property of logarithms, we can rewrite the equation as:
ln y = ln [(2x+1)^(1/6)/(x+4)^(1/9) *C^(1/3)]
Now, we can simplify further by using the property of exponential functions, which states that e^ln(x) = x. Applying this property to both sides of the equation, we get:
y = e^(ln y) = e^[ln [(2x+1)^(1/6)/(x+4)^(1/9) *C^(1/3)]]
Simplifying further, we obtain:
y = C^(1/3)* (2x+1)^(1/6) /(x+4)^(1/9)
Therefore, y is expressed as a function of x as:
y = C^(1/3)* (2x+1)^(1/6) /(x+4)^(1/9)