Question

Solve the given integral equation or integro-differential equation for y(t).

y(t)+4 ∫₀ᵗ(t-v) y(v) d v=16 t²

Answer 1

The solution to the given integral equation is
`\( y(t) = 4t^2 - 8t^3 + 6t^4 - (4)/(3)t^5 \)`.

Step-by-step explanation:

In solving this integral equation, we start by differentiating both sides with respect to t to convert it into a differential equation. After solving the resulting differential equation, we integrate back to obtain the solution for y(t). The initial steps involve computing the derivatives and integrals of the given functions, and then applying the boundary conditions, if provided. The final solution is a polynomial expression in terms of t, which satisfies the given integral equation.

The integral equation provided is a linear integro-differential equation with a quadratic function on the right-hand side. By differentiating both sides with respect to t, we transform it into a linear ordinary differential equation. Solving the resulting differential equation involves finding the particular solution and applying the initial conditions. After integrating back, we obtain the expression for y(t) a polynomial function with terms up to the fifth degree.

In summary, the solution
`\( y(t) = 4t^2 - 8t^3 + 6t^4 - (4)/(3)t^5 \)` is derived through a systematic process of differentiating, solving, and integrating to obtain the solution to the given integro-differential equation. The polynomial expression represents the function y(t) that satisfies the given equation for the specified conditions.