can someone makethis as hand written please :

Next, I solved the differential equation to arrive at the formula y(t) = M / (1 + A e^(-kt)), in these steps:

Start with the differential equation: y'(t) = ky(t)(1 - y(t)/M).

Separate the variables and integrate: dy(t) / (y(t)(M - y(t))) = k dt.

Integrating both sides using partial fractions decomposition, we get: dy(t) / (y(t)(M - y(t))) = (A/M)dy(t)/y(t) + (B/(M - y(t)))dy(t)/(M - y(t)), where A and B are constants to be determined.

Combine the fractions and integrate: ln|y(t)/(M - y(t))| = (A/M)ln|y(t)| - (B/M)ln|(M - y(t))|.

Simplifying the expression we get: ln|y(t)/(M - y(t))| = ln|y(t)^A / (M - y(t))^B|.

Therefore, we can write: y(t) / (M - y(t)) = y(t)^A / (M - y(t))^B.

Solve for A and B using the initial condition y(0) = y0.

At t=0, y(0) = y0, so we have: y(0) / (M - y(0)) = y(0)^A / (M - y(0))^(A+1).

Solving for B we get: B = A + 1.

Substituting this in the previous equation, we have: y(0) / (M - y(0)) = y(0)^A / (M - y(0))^(A+1).

Solving for A, we get: A = ln(y(0)/(M - y(0))) / ln(y(0)/M - 1).

Substitute the values of A and B back into the expression for y(t): y(t) / (M - y(t)) = y(t)^A / (M - y(t))^(A+1).

Simplifying this expression, we get: y(t) = M y(0) e^(kM t) / (y(0) + (M - y(0)) e^(kM t)), where k = ln(9) / (58M).