Question

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

First order differential equations
Solve differential equation $$L'(x)=k(x+b)(L-a)$$

$$(dL)/(L-a)= k(x+b)dx$$
$$(dL)/(L-a)= (kx+bk)dx$$
$$\int (dL)/(L-a)= \int (kx+bk)dx$$ $$\ln(L-a)= (kx^2)/(2+bkx+c)$$ (because $$\int (1/(x+a)dx=\ln(x+a))$$ and $$\int x^n dx= x^{(n+1)/(n+1)}$$ where c is the constant of integration
So, the solution of the differential equation $$L'(x) = k(x+b)(L-a)$$ will be $$\ln(L-a)= (kx^2)/(2 + bkx + c)$$