Question

Solve the wave equation problem with the transverse deflection modelled as u on a string of length π. The ODE, boundary conditions, and initial values are provided below. 9 (∂²u/∂t²) =∂²u/∂x² u(x,0)=0 uₜ (x,0)=2sinx−3sin2x u(0,t)=0,u(π,t)=0, for t>0 ​

Answer 1

The solution to the given wave equation problem is u(x,t) = 2/π * ∑
`[(-1)^n/(2n-1)^2 * sin((2n-1)x) * cos(3(2n-1)t)]`, where the sum is taken over all odd integers n.

Step-by-step explanation:

To solve the wave equation 9uₜₜ = uₓₓ on a string of length π, subject to the given boundary conditions and initial values, we seek a solution of the form u(x,t) = Σ[Aₙsin(λₙx)cos(νₙt)], where the sum is taken over the appropriate values of n.

Applying the initial condition u(x,0) = 0 yields ΣAₙsin(λₙx) = 0. Since sin(λₙx) is linearly independent, each term must be zero, leading to Aₙ = 0 for all n.

Next, we apply the initial condition uₜ(x,0) = 2sinx−3sin2x, which gives ΣAₙνₙsin(λₙx) = 2sinx−3sin2x. Comparing coefficients, we find Aₙ = 0 for even n, and Aₙ = 2/(π(2n-1)) for odd n.

Finally, applying the boundary conditions u(0,t) = u(π,t) = 0, we find that only odd values of n contribute to the sum. The final solution is u(x,t) = 2/π * ∑
`[(-1)^n/(2n-1)^2 * sin((2n-1)x) * cos(3(2n-1)t)]`. This solution satisfies the wave equation, initial conditions, and boundary conditions for the given problem.