Find solution to the differential equation

Question

$\frac{dB}{dx}+2B=50$

$B(1) = 50$

I tried separating the variables but that didn't work, and without separating the variable I'm not sure what to do.

Answer 1

separating the variable works:

$$ \frac {dB}{dx} = 50 - 2B \\ \int \frac{dB}{25 - B} = \int 2dx + C, C\in\Bbb R \\ -\log |25 - B| = 2x + C, C\in\Bbb R $$ yields the general solution: $$ B = 25 + K\exp (-2x), K\in\Bbb R $$ and with the initial condition: $$ B = 25 (1+ \exp (2(1-x))) $$

Answer 2

you can solve this O.D.E as follows $$\frac{dB}{dx}+2B=0$$ $$m+2=0$$ $$m=-2$$ $$y_c=C_1e^{-2x}$$ to find the particular solution $$y_p=A$$ then the $A=25$ hence $$y=C_1e^{-2x}+25$$