Question

Consider the following differential equation.

(x² − 4)

dy

dx

+ 4y = (x + 2)²

^{Consider the following differential equation. dy (x2 - 4) dx + 4y = (x + 2)2 Find the coefficient function P(x) when the given differential equation is written in the standard form dy dx + P(x)y = f(x). 4 P(x) = (x2 – 4) Find the integrating factor for the differential equation. SP(x) dx 1 Find the general solution of the given differential equation. y(x) = Give the largest interval over which the general solution is defined. (Think about the implications of any singular points. Enter your answer using interval notation.) Determine whether there are any transient terms in the general solution. (Enter the transient terms as a comma-separated list; if there are none, enter NONE.) Need Help? Read It Watch It Talk to a Tutor}

Answer 1

In + 4y = (x+2) 2 xy y = (x+292 OY) dät (x-4) dy dy nay comparing (1) with the differential equation dy + P (n) y = f(n), P(x) = 4 ày Soda . Integrating factor (1.f) - ě an e de ! e don 4 x 22 - e

4.22locatz 11 log (x^2 a+2 e x-2 xt 2 Multiplying (1) by If, we get 2 2-2 + X-2 dy atz da y 2-4 (x + 2)2 2-2 2-4 nt2 ont 2 y = d त [ y. H-2 +2 x + 2) (x-2) (x+2)(x-2) n-2 n+2 -da on. integrating, y. 2-2 nt2 xt C where cis constant. or you @2)x4 (32) at 12 Largest interval over which the general Solution is defined is (-2,-2) U(+2, 2) U (2,00) Transient term is comi2).