Let's solve the given differential equation and find the limit
y(x).The differential equation given is:
xy′+y−ex=0 This is a first-order linear differential equation. We can rewrite it in the form:
y′+‌y=‌We will solve this using the integrating factor method. The integrating factor (IF) is given by:
µ(x)=e∫‌‌dx=eln|x|=|x| Since we are not given a specific interval for
x, we assume
x is positive:
µ(x)=xMultiplying both sides of the differential equation by the integrating factor:
&xy′+y=ex⇒xy′+y=ex⇒x(y′+‌y)=ex&x⋅y′+y=ex Recognize that the left side is the derivative of
xy :
‌(xy)=exIntegrate both sides with respect to x :
xy=∫ex‌dx=ex+CSo we have:
y=‌ Apply the initial condition
y(a)=b :
&b=‌&C=ab−ea Thus, the solution to the differential equation is:
y=‌Now, we need to determine:
y(x) Substituting
x=1 into the solution:
y(1)=‌=e+ab−eaTherefore:The correct answer is Option D
e+ab−ea