Math 465: Applied Mathematics I

Find a series solution to \( y' + xy = 1 + x \). Show that the homogeneous solution is \( e^{-x^2/2} \). Solution

Find two power series solutions to \( y'' - xy' - y = 0 \) about \(x_0 = 0\). Solution

(Equal roots) Solve \( xy'' + y' + y = 0 \) about \(x_0=0\). Solution

(Roots differ by an integer) Solve \( xy'' + y = 0 \) about \(x=0\). Find the first four non-zero terms. Solution

(Roots differ by an integer) Find the closed form solutions of \( xy'' + y = 0 \) about \(x=0\). Solution

(Roots differ by an integer) Solve \( x(x-1)y'' + 6x^2y' + 3y = 0 \) about \(x_0=0\). Find the first four non-zero terms. Solution

(Roots differ by an integer) Find the closed form solutions of \( x^2y'' + 2xy' + xy = 0 \) about \(x=0\). Solution

(Equal roots) Find closed forms of solutions of \( x^2y'' + 3xy' + (1+x)y = 0 \) about \(x_0=0\). Solution

Fourier Series example and an Even/Odd function problem

Derivation of Heat Conduction: Bar with Insulated Ends

Solve the following heat conduction problem using separation of variables: \[ 4u_{xx} = u_t, \quad 0 < x < 50, \, t > 0 \] \[ u(0,t) = 0, \quad u(50,t) = 0 \] \[ u(x,0) = 20, \quad 0 \le x \le 50 \] Solution

Wave equation: string set in motion from equilibrium with a given velocity

Find the eigenpairs of \( y'' + \lambda y = 0, \quad y'(0) = 0, \quad y'(1) + y(1) = 0 \)

Solve \( y'' + \lambda y = 0, \quad y(0) = 0, \quad y(\pi) + y'(\pi) = 0 \)

Find normalized eigenfunctions of \( y'' + \lambda y = 0, \quad y'(0) = 0, \quad y'(1) = 0 \)

Find normalized eigenfunctions of \( y'' + \lambda y = 0, \quad y(0) = 0, \quad y'(1) + y(1) = 0 \)

Eigenfunction expansion: \( y'' + 2y = -x, \quad y(0) = 0, \quad y'(1) = 0 \)

Eigenfunction expansion: \( y'' + 2y = -x, \quad y(0) = 0, \quad y'(1) + y(1) = 0 \)

Nonhomogeneous Heat Conduction Problem