If dy/dt = k/y and k is a nonzero constant, which of the following could be y?
A. y = √(2kt + 16)
B. y = kt + 5
C. y = √(kt + 16)
D. y = 5e^(kt)
E. y = √(2kt) + 4
I got E by finding the integral of y dy = k dt.
dy/dt = k/y
y dy = k dt
1/2 y^2 = kt + c
y^2 = 2kt + c
y = √(2kt+c)
E does not work, as you can easily see:
y = √(2kt)+4
dy/dt = 2k/(2√(2kt)) = k/√(2kt)
but that is not k/y = k/(√(2kt)+4)
To find which of the given options could represent y, you correctly set up the differential equation dy/dt = k/y and integrated both sides to get the equation ∫(1/y)dy = ∫k dt.
Now, let's check each option to see if it satisfies this equation.
A. y = √(2kt + 16)
To find the integral of (1/y)dy, use the substitution u = 2kt + 16, then du = 2k dt.
∫(1/y)dy = ∫(1/√u)du = √u.
After substituting back, we have √(2kt + 16).
This option does not satisfy the equation since it doesn't match the result of the integral.
B. y = kt + 5
The integral of (1/y)dy is ln|y|.
This option does not satisfy the equation since it's not equal to k t + C, where C is a constant.
C. y = √(kt + 16)
Using the same substitution as in option A, we have ∫(1/y)dy = ∫(1/√u)du = √u.
After substituting back, we have √(kt + 16).
This option satisfies the equation since it matches the result of the integral.
D. y = 5e^(kt)
The integral of (1/y)dy is ln|y|.
This option does not satisfy the equation since it's not equal to k t + C, where C is a constant.
E. y = √(2kt) + 4
Using the same substitution as in option A, we have ∫(1/y)dy = ∫(1/√u)du = √u.
After substituting back, we have √(2kt) + 4.
This option does not satisfy the equation since it doesn't match the result of the integral.
Therefore, the only option that satisfies the differential equation is C. y = √(kt + 16).