Question

Problem1 The behavior of a physical system can be described by the following first order differential equation: dy/dt=2y +t^2

Answer 1

Answer:

$y = \dfrac{t^3e^{2t}}{3}+Ce^{2t}$.

Step-by-step explanation:

Using first order linear differential equation:

$\dfrac{\mathrm{d} y}{\mathrm{d} t} = 2y + t^2$

$\frac{\mathrm{d} y}{\mathrm{d} t} - 2y =t^2$

finding integrating factor:

I.F = $e^{\int -2dt}$

I.F =$e^{-2t}$

now,

$y = \dfrac{1}{IF}(\int t^2dt+ c )$

$y = \dfrac{1}{e^{-2t}}(\int t^2dt+ c )$

$y = \dfrac{t^3e^{2t}}{3}+Ce^{2t}$

hence the solution is

$y = \dfrac{t^3e^{2t}}{3}+Ce^{2t}$