Help with Ordinary Differential Equation?

Question

Question was to find general solution of dy/dx = (4x+2y-1)^1/2

This is what I did: let z = 4x + 2y -1 therefore z'=4+2y' => y' = z'/2 - 2
Substitute to get z'/2 - 2 = z^(1/2) => z' = 2*z^(1/2) + 4 (A Bernouilli Equation) Dividing thru by z^(1/2) gives

z^(-1/2) * z' = 2 + 4*z^(-1/2)

Now say m = z^(-1/2)
=> m' = (-1/2)*z^(-3/2) => m' = (-1/2)*z^(-1/2)*z^-1
=> 2*m' = -m*z^-1
=> 2*m'/m = z^-1

Now integrating both sides gives

2 ln m = ln z + c

Does this look right? Where do I have to go from here?

Answer 1

At this line
z' = 2*z^(1/2) + 4
I think you might be much better off just separating hte variables;

dz / (z^1/2 + 2) = 2*dx
You can let u = z^1/2 to integrate the LHS

The problem with your last line
2*m'/m = z^-1
is that there is no dz. m' = dm/dx. So you can't integrate with respect to z. Usually the best test is the plug it in back into the original equation.

Answer 2

z' = 2*z^(1/2) + 4 is not Bernoulli.

However, it is separable.