Final answer:
To solve the given differential equation, we separate variables and integrate both sides of the equation.
Step-by-step explanation:
To solve the differential equation 4y" - 4y'y = 0, we can first separate the variables. Then, we can integrate each side to find the general solution.
Let's start by separating the variables. We can write the differential equation as:
4y" - 4y'y = 0
4y" = 4y'y
Next, we divide both sides by 4y'y:
y"/y' = 1/y
Now, we can integrate both sides with respect to x:
∫(y"/y')dx = ∫(1/y)dx
Using the properties of logarithms, we have:
ln|y'| = ln|y| + C1
where C1 is the constant of integration.
Finally, we can solve for y' by exponentiating both sides:
y' = y * e^C1
This is a separable differential equation that can be solved by separating the variables and integrating.