How to solve the following first order differential equation?
$\dfrac{dy}{dt}+kty(t) = \dfrac{\sin(\pi t/10)}{\pi}$
How to solve the following first order differential equation?
$\dfrac{dy}{dt}+kty(t) = \dfrac{\sin(\pi t/10)}{\pi}$
First note that $\dfrac{dy}{dt} + kty = e^{-kt^2/2} \left(e^{kt^2/2} \dfrac{dy}{dt} + e^{kt^2/2}kty \right) = e^{-kt^2/2} \dfrac{d}{dt}\left(e^{kt^2/2} y\right) = \dfrac{\sin \left(\dfrac{\pi t}{10}\right)}{10}$ Hence, we have that $\dfrac{d}{dt}\left(e^{kt^2/2} y\right) = e^{kt^2/2} \dfrac{\sin \left(\dfrac{\pi t}{10}\right)}{10}$ Hence, $e^{kt^2/2}y(t) = y(0) + \dfrac1{10} \int_0^t e^{kx^2/2} \sin \left(\dfrac{\pi x}{10}\right) dx$ $y(t) = y(0)e^{-kt^2/2} + \dfrac{e^{-kt^2/2}}{10} \int_0^t e^{kx^2/2} \sin \left(\dfrac{\pi x}{10}\right) dx$
This is a first order linear differential equation. You can solve it by using the integrating factor method. Here is a reference: http://www.sosmath.com/diffeq/first/lineareq/lineareq.html