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

Problem Statement
Find and plot the solution for the homogeneous L2-ODE-CC
 * $$y''(x) - 10y'(x) + 25y(x)=0\!$$

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

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 $$y(x)=c_1e^{\lambda x}+c_2xe^{\lambda x}\!$$

First initial condition

$$y(0)=1\!$$

$$y(0)=c_1e^{5*0}+c_2*0*e^{5*0}=1\!$$

$$c_1=1\!$$

Second initial condition

$$y'(0)=0\!$$

$$\frac{d}{dx}y(x)=y'(x)=5e^{5x}+c_2e^{5x}(5x+1)\!$$

$$y'(0)=5e^{5*0}+c_2e^{5*0}(5*0+1)=0\!$$

$$5+c_2=0\!$$

$$c_2=-5\!$$

The solution to our L2-ODE-CC is

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

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



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