solve the differential equation?

dy/dx + a*y = exp(x), a ≠ -1

This is a first-order linear equation that we can solve using an intgrating factor.

For equations of the form:

dy/dx + a(x)*y = b(x),

the integrating factor, p(x) is given by:

p(x) = exp(INTEGRAL of {a(x) dx}) [we don't have to worry about the constant of integration at this step because it turns out it can always be combined with the constant arising from the next integration]

and the solution is given by:

y(x) = (1/p(x))*INTEGRAL of {p(x) * b(x) dx}

In this case,

p(x) = exp(INTEGRAL of {a dx}) = exp(a*x)

So:

y(x) = exp(-a*x)*INTEGRAL of {exp(a*x) * exp(x) dx}

y(x) = exp(-a*x)*INTEGRAL of {exp(a*x + x) dx}

y(x) = exp(-a*x)*INTEGRAL of {exp((a+1)*x) dx}

y(x) = exp(-a*x)*[((1/(a+1))*exp((a+1)*x) + c]

where c is the constant of integration.

y(x) = exp(x)/(a+1) + c*exp(-a*x)