Ordinary differential equations
Solve the ode y’’ – y’ +4y = 0 as a system of first order odes.
Use the method of undetermined coefficients to find all solutions of y''(x) -y(x) = 5e^x sin x .
Matrix relative to standard basis
Let D: P5 |-> P5, Where D = d /dx . Find the matrix of the D relative to the standard basis, { 1,x, x^2,x^3,x^4,x^5}.
Legendre's differential equation
Legendre's differential equation, i.e. , (1-x^2)y''(x) - 2xy'(x) + n(n+1)y(x)=0. Find all solutions to Legendre's differential equation assuming solutions of the form y(x) = x^r.
Legendre's differential equation
Legendre's differential equation, i.e. , (1-x^2)y''(x) - 2xy'(x) + n(n+1)y(x)=0. Find all solutions to Legendre's differential equation assuming solutions of the form y(x) = P r (x)
Bessel functions and Legendre polynomials
1).The Bessel functions Jp satisfy the equations (see attached file) . Use this to show that Bessel functions satisfy Bessel's differential equation. 2) The legendre polynomials satisfy... (see attached) where Pn(x) is the Legendre polynomial of degree n. Show that the Legendre satisfy Legendre's differential equation, (see atta ...continues
Fibonacci Sequence : Diagonalization of Iteration Matrix
The Fibonacci sequence is defined recursively as f_(n+1) = f_n + f_(n-1) . Obtain a closed form formulas for f_n using the iteration matrix (0,1,1,1) applied to the vector .. (see attached file). Consider a similar transform to diagonalize the iteration matrix.
Laplace Transform and Inverse Laplace Transform
1. L [te^2t sin 3t] 2. L^-1 [(2s + 3)/(s^2 + 2s - 8)] Please see the attached file for the fully formatted problems.
Please work on problem #2 L^-1[(2S + 3)/(S^2 + 2S - 8)]. Please see the attached file for the fully formatted problem.
Initial-Value Problems and Laplace Transforms (5 Problems)
Please show all steps for the following problem. Section 7.5 # 4,8,16,20,26 Please see the attached file for the fully formatted problems.
