University of Florida/Egm4313/s12.team11.imponenti/R2.2

Report 2, Problem 2

Problem Statement

Find and plot the solution for the homogeneous L2-ODE-CC

$y^{″}(x)-10y^{\prime }(x)+25y(x)=0$

with initial conditions $y(0)=1$ ,and $y^{\prime }(0)=0$

Characteristic Equation

${\lambda }^2-10\lambda +25=0$

$(\lambda -5)(\lambda -5)=0$

$\lambda =5$

Homogeneous Solution

The solution to a L2-ODE-CC with real double root is given by

$y(x)=c_1{e}^{\lambda x}+c_{2}x{e}^{\lambda x}$

First initial condition

$y(0)=1$

$y(0)=c_1{e}^{5*0}+c_2*0*e^{5*0}=1$

$c_1=1$

Second initial condition

$y^{\prime }(0)=0$

$\frac{d}{dx}y(x)=y^{\prime }(x)=5e^{5x}+c_2{e}^{5x}(5x+1)$

$y^{\prime }(0)=5e^{5*0}+c_2{e}^{5*0}(5*0+1)=0$

$5+c_2=0$

$c_2=-5$

The solution to our L2-ODE-CC is

                       $y(x)=e^{5x}(1-5x)$

Plot

$y(x)=e^{5x}(1-5x)$

File:PlotR2 2.jpg

Egm4313.s12.team11.imponenti 00:30, 8 February 2012 (UTC)