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

Problem Statement

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

${\displaystyle y''(x)-10y'(x)+25y(x)=0\!}$

with initial conditions ${\displaystyle y(0)=1\!}$ ,and ${\displaystyle y'(0)=0\!}$

Characteristic Equation

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

${\displaystyle (\lambda -5)(\lambda -5)=0\!}$

${\displaystyle \lambda =5\!}$

Homogeneous Solution

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

${\displaystyle y(x)=c_{1}e^{\lambda x}+c_{2}xe^{\lambda x}\!}$

First initial condition

${\displaystyle y(0)=1\!}$

${\displaystyle y(0)=c_{1}e^{5*0}+c_{2}*0*e^{5*0}=1\!}$

${\displaystyle c_{1}=1\!}$

Second initial condition

${\displaystyle y'(0)=0\!}$

${\displaystyle {\frac {d}{dx}}y(x)=y'(x)=5e^{5x}+c_{2}e^{5x}(5x+1)\!}$

${\displaystyle y'(0)=5e^{5*0}+c_{2}e^{5*0}(5*0+1)=0\!}$

${\displaystyle 5+c_{2}=0\!}$

${\displaystyle c_{2}=-5\!}$

The solution to our L2-ODE-CC is

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

Plot

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

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