How do I solve the equation dy/dt = 2y - 10?

Jan 31, 2015

You can use a technique known as Separation of Variables.
Take all the $y$ to one side and the $t$ on the other...
You get:

$\frac{\mathrm{dy}}{2 y - 10} = \mathrm{dt}$

Now you can integrate both sides with respect to the correspondent variables:

$\int \frac{1}{2 y - 10} \mathrm{dy} = \int \mathrm{dt}$
$\int \frac{1}{2 \left(y - 5\right)} \mathrm{dy} = \int \mathrm{dt}$

And finally
$\frac{1}{2} \ln \left(y - 5\right) = t + c$

Now you can express $y$ as:
$\ln \left(y - 5\right) = 2 t + c$
$y - 5 = {c}_{1} {e}^{2 t}$ where ${c}_{1} = {e}^{c}$
$y = {c}_{1} {e}^{2 t} + 5$

You can substitute back to check your result (calculating $\frac{\mathrm{dy}}{\mathrm{dt}}$) remembering that now it is: $y = {c}_{1} {e}^{2 t} + 5$