An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation. [Hint: Use an initial condition obtained from the integral equation.] y(x) = 2 + x [t − ty(t)] dt 2

Step-by-step explanation:

We have that:

     (Equation 1)

To resolve this integral equation, we need to use the second Fundamental Theorem of Calculus, which says:

So, we need to differentiate both sides of equation 1 with respect to x:

We know that the derivate for a constant value is zero. And,

        (Equation 2)

Using the second Fundamental Theorem of Calculus we know that:

So, we need to replace those equations in equation 2, and we obtain:

      (Equation 3)

Now, we are going to resolve the equation 3 as a normal equation. So, we need to joint the same variables. I mean, variable y on one side and variable x on other side, as follows:

And, we integrate each side of the equation to obtain:

     (Equation 4)

Now, we need to find the value of the constant C. And we know that we can find one point of the equation, replacing x=2 in equation 1, because the integral becomes zero, so:

And, we replace the value of y when x=2 in equation 4 and we obtain,

So, Equation 4 is:

        (Equation 4')

Now, we need to clear the y variable from Equation 4', (we are going to asumme that y-1>0),