anonymous 4 years ago 1. Find the general solution (or as close as you can come to it) for the following differential equations, using separation of variables. (a) dy/dt = 2y − 1 (b) dy/dt = t^2y^3

1. dumbcow

$\frac{dy}{dt} -2y = -1$ find integrating factor $e^{\int\limits_{}^{}-2} = e^{-2t}$ $\frac{dy}{dt}e^{-2t} -2e^{-2t}y = -e^{-2t}$ $(e^{-2t}y)\frac{d}{dt} = -e^{-2t}$ integrate both sides with respect to t $e^{-2t}y = \frac{1}{2}e^{-2t} + C$ $y = \frac{1}{2} +Ce^{2t}$

2. dumbcow

$\frac{dy}{dt} = t^{2}y^{3}$ separate variables $\frac{dy}{y^{3}} = t^{2} dt$ integrate both sides $-\frac{1}{2y^{2}} = \frac{t^{3}}{3}+C$ $2y^{2} = \frac{3}{C-t^{3}}$ $y = \sqrt{\frac{3}{C-2t^{3}}}$