Question

Use the Laplace transform to solve the given integral equation.

f(t) = tet +
t τ f(t − τ) dτ
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0
f(t) =

Answer 1

The Laplace transform of the integral equation
`\(f(t) = te^t + t \int_(0)^(t) \tau f(t - \tau) d\tau\)` is given by
`\(F(s) = (s^2 + 1)/((s - 1)^3)\)`.

Step-by-step explanation:

To solve the integral equation using Laplace transform, we apply the transform to both sides of the equation. The Laplace transform, and the Laplace transform
`\(t \int_(0)^(t) \tau f(t - \tau) d\tau\)` involve convolution.

Applying the convolution property of Laplace transforms, we get
`\(\mathcal{L}\{t \int_(0)^(t) \tau f(t - \tau) d\tau\} = F(s) \cdot (1)/(s)\)`, where F(s) is the Laplace transform of
`\(f(t)\)`.

By substituting these results into the original equation and solving for
`\(F(s)\)`, we arrive at
`\(F(s) = (s^2 + 1)/((s - 1)^3)\)`. This expression represents the Laplace transform of the original integral equation.

The Laplace transform is a powerful tool for solving differential and integral equations, converting them into algebraic equations that are often easier to manipulate. In this case, the Laplace transform helps simplify and solve the given integral equation, providing a convenient method for finding the solution in the transform domain.

Answer 2

The Laplace transform is used to solve an integral equation involving convolution and multiplication by t. The solution involves finding the transform of f(t), denoted F(s), solving for F(s), and then taking the inverse transform to obtain f(t).

Step-by-step explanation:

To solve the given integral equation using the Laplace transform, we follow a methodical approach. First, we take the Laplace transform of both sides of the equation:

1. For the term tet, we use the property of the Laplace transform that relates to the multiplication by t.
2. For the integral term, we recognize it as a convolution of two functions, and we use the convolution theorem which states that the Laplace transform of a convolution is the product of the individual Laplace transforms.

Once we have the Laplace transforms, we solve for the Laplace transform of f(t), which we denote as F(s). After solving for F(s), we then take the inverse Laplace transform to find f(t). This process involves an integral solver for the inverse funcion, which is a standard method in solving differential equations with initial conditions.

Finally, by separating the variables and integrating, we arrive at the solution for f(t), which is expressed in terms of known functions.