Solve each of the following initial value problems: y' + y = e^x, y(0) = 1/2

Question

Solve each of the following initial value problems:

\(y'+y=e^x, y(0)=\frac{1}{2}\)

Answer 1

This is a first order linear differential equation of the form

The integrating factor (I.F) of this differential equation is,

Hence, the solution of the differential equation is,

However, when x = 0, we have y = \(\frac{1}{2}\)

By substituting the value of c in the equation for y, we get

Thus, the solution of the given initial value problem is \(y=\frac{e^x}{2}\)