Consider the differential equationdydx−(2x+3)y=(3x+2)e2x.dydx−(2x+3)y=(3x+2)e2x.  a) It is given that if both sides of the differential equation are multiplied by a function  ϕ(x)ϕ(x) ,  the differential equation will become ddx(ϕ(x)y)=ϕ(x)(3x+2)e2xddx(ϕ(x)y)=ϕ(x)(3x+2)e2x . Find  ϕ(x)ϕ(x)  and write your answer in the box below. ϕ(x)=ϕ(x)=          b) Setting the initial condition  y(1)=0y(1)=0 , what is the largest domain  II  for the solution  yy ? I=I=

a) The function ϕ(x) is called an integrating factor. It is used to transform a first-order linear differential equation into an exact differential equation. The integrating factor is usually of the form e^(∫P(x)dx), where P(x) is the coefficient of y in the differential equation. In this case, P(x) = -(2x + 3). So, we have:

ϕ(x) = e^(∫-(2x + 3)dx) = e^(-x^2 - 3x)

b) The largest domain I for the solution y is the set of all x for which the function ϕ(x)y and its derivative are defined. Since ϕ(x) = e^(-x^2 - 3x) is defined for all real numbers, and y(1) = 0 is given, the largest domain I is the set of all real numbers, or I = (-∞, ∞).