Question

Use D'Alembert's method to find the solution of the wave equation ∂t ² /∂² u​ =4 ∂x² /∂² u​ for t>0 and all x∈R with initial conditions u(x,0)=sinx and ∂t∂u(x,0)​ =e −x/4 .

Answer 1

D'Alembert's method is a technique used to find solutions to partial differential equations, such as the wave equation. We can then find the general solution by finding appropriate functions F and G.

Step-by-step explanation:

D'Alembert's method is a technique used to find solutions to partial differential equations, such as the wave equation. In this case, we have the wave equation ∂t² / ∂x² u = 4 ∂x² / ∂x² u.

To solve this equation using D'Alembert's method, we can introduce two new variables, ξ and η, defined as follows:

ξ = x - t, η = x + t

Using these variables, the wave equation can be rewritten as:

4(∂² / ∂ξ² - ∂² / ∂η²) u = 0

This is a homogeneous equation in ξ and η.

We can solve this equation by assuming a solution of the form u(ξ, η) = F(ξ) + G(η), where F and G are arbitrary functions.

We can then find the general solution by finding appropriate functions F and G.