Question

Find the solution of the given differential equation and then find the particular solution for which a point (x,y) is given:

dy/dx = (2-x) (x+1); (x,y) = (-1,3)

Answer 1

$y'\:=\left(2-x\right)\left(x+1\right)\\ \mathrm{If\quad }f'\left(x\right)=g\left(x\right)\mathrm{\quad then\quad }f\left(x\right)=\int g\left(x\right)dx\\ y=\int \left(2-x\right)\left(x+1\right)dx\\ y=\frac{x^2}{2}-\frac{x^3}{3}+2x+c_1\\ y=\frac{x^2}{2}-\frac{x^3}{3}+2x+\frac{25}{6}$