Integral Equations
In Exercises 7–12, write an equivalen...

Question

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = 1 + ∫₀ ͯ y(t) dt

Accepted Answer

Welcome back everyone. Let y of x be equal to the integral from pi up to x of cosinet divided by t dt. Find an equivalent first order differential equation and initial condition for y. For this problem, let's begin by finding the differential equation, and we can do that by taking the derivative of both sides. So dyy divided by dx, which is the derivative of y of x, is going to be the derivative d divided by dx of the integral from pi up to x of cosinet divided by t dt. And to find the derivative of this integral we can apply the fundamental theorem of calculus part one. Because the lower limit of integration is a constant and the upper limit of integration is x, so according to the theorem, the value is going to be our function cosinet divided by t evaluated at x. In other words, t is going to become x, and this gives us cosine x divided by x. So that's how we get the required differential equation. It is d y divided by. D x is equal to cosine x divided by x, and all that we have to do is identify the initial condition. So the easiest way is to evaluate why at the lower limit of integration, why at pi. It is going to be equal to the integral from pi up topi of cosinet divided by t dt. And the reason why this works is because whenever the lower limit of integration is equal to the upper limit of integration, the integral is equal to zero. This is how we get the initial condition y at pi is equal to 0. That's our final answer, and thank you for watching.

Integral EquationsIn Exercises 7–12, write an equivalent first-order differential equation and initial condition for y.y = 1 + ∫₀ ͯ y(t) dt

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Integral Equations
In Exercises 7–12, write an equivalent first-order differential equation
and initial condition for y.

y = 1 + ∫₀ ͯ y(t) dt