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

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

2020-12-18
$$(dL)/(L-a)= k(x+b)dx$$
$$(dL)/(L-a)= (kx+bk)dx$$
Now integrating both sides
$$int (dL)/(L-a)= int (kx+bk)dx$$ $$ln(L-a)= (kx^2)/(2+bkx+c)$$ (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 $$ln(L-a)= (kx^2)/(2 + bkx + c)$$

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