1.The ODE X"+ kX = 0 has different types solution depending on sign of k. We consider the three possible cases separately. k=0: X"=0 so that X(x)=Ao+Box, X'(0)=0 implies Bo=0 so that X(x)=Ao. Finally, X(1)=0 implies Ao=0 and there is only trivial solution X=0. The matrix has determinant e^mu + e^-mu= 2cosh.mu, which is ...continues
Wave equation : Solve the wave equation using d'Alembert's method.
Solve the wave equation using d'Alembert's method. Please see the attached file for the fully formatted problems.
Wave equations : Solve the boundary value problem using separation of variables.
Solve the boundary value problem using separation of variables. Please see the attached file for the fully formatted problems.
1-D Heat Equations : Solve the Boundary Value Problem
Please see the attached file for the fully formatted problems.
1. When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Questi ...continues
Background Information: A simple pendulum, such as a rock hanging from a piece of string or the inside of a grandfather clock, consists of a mass (the rock) and a support (the piece of string). When the mass is moved a small distance away from its equilibrium point (the bottom of the arc), the mass will swing back and forth ...continues
Boundary value problem for Laplace's equation (Dirichlet Problem)
Boundary value problem for Laplace's equation (Dirichlet Problem). See attached file for full problem description.
Partial Differential Equation Solution
1. Consider the partial differential equation: q(x) u/t = /x (p(x) u/x) (a) Confirm that U(x) = A + (B-A)(integral from 0 to x (p(s)^-1 ds))/(integral from 0 to L (p(s)^-1 ds)) Is a solution of the partial differential equation (b) Confirm that if u=U+v then v indeed satisfi ...continues
Sturm-Liouville Problems : Eigenvalues, Eigenfunctions and Square Norm
1. find the eigenvalues and eigenfunctions of the boundary- value problem y''+λ y = 0 , y(0) = 0 , y(π /4) = 0 ( n = pi) 2. find the eigenvalues and eigenfunctions of the boundary- value problem y''+(λ+1) y = 0 , y'(0) = 0 , y'(1) = 0 3. find the sqaure norm of the following problem along with t ...continues
Fourier-Legendre Expansion of F(θ) = 1 - cos 2θ
Number 18. Please don't do the CAS nor the plot. Please see the attached file for the fully formatted problems.
