Math 465: Applied Mathematics I

Find a series solution to the differential equation \( y'+xy=1+x\). Also show that the homogeneous solution is equal to \( \texttt{e}^{-\frac{x^2}{2}}\).    SOLUTION

Find two power series solutions to the given differential equations about  \(x_0 = 0. \qquad y''-xy'-y=0\)     SOLUTION  

(equal roots) Find two series solutions of the ode  \(xy''+y'+y=0 \) about \(x_0=0\).    SOLUTION

(roots differ by an inter) Solve the differential equation \(xy''+y=0 \) abut \(x=0\). Find the first four non-zero terms.  SOLUTION

(roots differ by an inter) Find the closed form solutions of \(xy''+y=0\) about \(x=0.\)   SOLUTION

(roots differ by an inter) Solve the differential equation \(x(x-1)y''+6x^2y'+3y=0 \) abut \(x_0=0\). Find the first four non-zero terms.   SOLUTION

(roots differ by an inter) Find the closed form solutions of \(x^2y''+2xy'+xy=0\) about \(x=0.\)   SOLUTION

(equal roots) Find closed forms of solutions of the ode  \(x^2y''+3xy'+(1+x)y=0 \) about \(x_0=0\).   SOLUTION

Fourier Series example and an Even Odd function problem

Derive the solution of Heat Conduction: Bar with Insulated Ends.

Solve the following heat conduction problem using separation of variables.  SOLUTION

\[ 4u_{xx}=u_t, \quad 0<x<50, \, t>0\]

\[ u(0,t)=0, \: u(50,t)=0 \]

\[ u(x,0)=20,  \: 0 \le x \le 50\]

Derive the solution:  wave equation problem: string is set in motion from its equilibrium position with a given velocity.

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

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

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

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

Solve the given problem by means of an eigenfunction expansion \(\quad y''+ 2 y = -x, \quad y(0)=0, \: y'(1) = 0\)

Solve the given problem by means of an eigenfunction expansion \(\quad y''+ 2 y = -x, \quad y(0)=0, \:  y'(1) + y(1) = 0\)

Nonhomogeneous Heat Conduction Problem

- page under construction!