I would really appreciate it if someone could help me.
Solve the differential equation by separation of variables.
dy/dx = (y+5)(x+2)
Thanks.
We have to solve on some open interval,
$\displaystyle y'=(y+5)(x+2)$.
We want to divide by $\displaystyle y+5$.
But we have to consider the case where $\displaystyle y+5=0$ in that case $\displaystyle y=-5$.
By substitution it does solve the differencial equation.
Second case consider $\displaystyle y+5\not =0$*
And divide through,
$\displaystyle \frac{y'}{y+5}=x+2$
$\displaystyle \int \frac{y'}{y+5} dx=\int x+2 dx$
$\displaystyle \ln |y+5|=\frac{1}{2}x^2+2x+C_1$
$\displaystyle |y+5|=\exp (1/2x^2+2x+C_1)=C\exp(1/2x^2+2x), C>0$
Thus,
$\displaystyle y+5=\pm C \exp(1/2 x^2+2x)=C\exp (1/2 x^2+2x), C\not =0$
$\displaystyle y=-5+C\exp (1/2x^2+2x), C\not =0$
These are the necessary solutions, check them to show they all work.
*)Note there is a case where $\displaystyle y+5=0$ for some point in the open interval and non-zero in some point. If that where the case then since the function is continous there is an open interval where it is non-zero and hence the solution we have above. But then there is no way that the function will connect will the zero point for that will lead to non-continuity and hence non-differenciability. But that cannot be the case because we assume $\displaystyle y$ is differenciable on the open interval. Thus that cannot be the case.