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

Question

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

dumbcow · Answer

$$\frac{dy}{dt} -2y = -1$$
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
$$e^{-2t}y = \frac{1}{2}e^{-2t} + C$$
$$y = \frac{1}{2} +Ce^{2t}$$

dumbcow · Answer

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