Solve the differential equation L'(x) = k (x+a) (L-b)?

You have:

L'(x) = dL/dx = k*(x+a)*(L-b)

Separate the variables:

dL/(L -b) = k*(x+a) dx

Integrate both sides:

ln(L - b) = k*((x^2)/2 + a*x) + c

where c is the combined constant of integration.

L = b + exp( k*((x^2)/2 + a*x) + c)

Note that exp( n + n) = exp(n)*exp(m), so we can write this as:

L(x) = b + C*exp(k*((x^2)/2 + a*x))

where C = exp(c) is just another way of writing the constant.

Take L'=dy/dx
dy/dx= k(x+a)(y-b)
dy/y-b=k(x+a) dx
log(y-b)=kx^2 + kax+ C

L'(x) = k (x+a) (L-b)
L'/(L-b) = kx + ka
Integrate it.
ln (L-b) = 1/2 kx^2 + kax +b
L-b= e^(1/2 kx^2 + kax + b)
L = b +e^(1/2 kx^2 + kax + b)