University of Florida/Egm4313/s12.team11.gooding/R2/2.1

Problem Statement
Given the two roots and the initial conditions: $$\lambda_{1}=-2,\lambda_{2}=5 \!$$ $$y(0)=1,y'(0)=0\!$$ Find the non-homogeneous L2-ODE-CC in standard form and the solution in terms of the initial conditions and the general excitation $$r(x)\!$$. Consider no excitation: $$ r(x)=0\!$$ Plot the solution

Characteristic Equation:
$$ (\lambda-\lambda_1)(\lambda-\lambda_2)=0 \!$$ $$ (\lambda+2)(\lambda-5)=\lambda^2 +2\lambda-5\lambda-10=0\!$$ $$\lambda^2 -3\lambda-10=0\!$$

Non-Homogeneous L2-ODE-CC
$$y''-3y'-10=r(x)\!$$

Homogeneous Solution:
$$ y_{h}(x)=c_1e^{-2x}+c_2e^{5x}\!$$ $$ y(x)=c_1e^{-2x}+c_2e^{5x}+y_p(x)\!$$ Since there is no excitation, $$y_p(x)=0\!$$ $$ y(x)=c_1e^{-2x}+c_2e^{5x}\!$$

Substituting the given initial conditions:
$$ y(0)=1\!$$ $$1=c_1+c_2\!$$ $$y'(0)=0\!$$ $$0=-2c_1+5c_2\!$$ Solving these two equations for $$ c_1\!$$ and $$ c_2\!$$ yields: $$ c_1=5/4, c_2=-1/4 \!$$

Final Solution
$$ y(x)=(5/4)e^{-2x}-(1/4)e^{5x}\!$$

Problem Statement
Generate 3 non-standard (and non-homogeneous) L2-ODE-CC that admit the 2 values in (3a) p.3-7 as the 2 roots of the corresponding characteristic equation.

Solutions
$$ 2(\lambda+2)(\lambda-5)=2\lambda^2 +4\lambda-10\lambda-20=0\!$$ $$2\lambda^2 -6\lambda-20=0\!$$ $$ 3(\lambda+2)(\lambda-5)=3\lambda^2 +6\lambda-15\lambda-30=0\!$$ $$3\lambda^2 -9\lambda-30=0\!$$ $$ 4(\lambda+2)(\lambda-5)=4\lambda^2 +8\lambda-20\lambda-40=0\!$$ $$4\lambda^2 -12\lambda-40=0\!$$ --Egm4313.s12.team11.gooding 02:01, 7 February 2012 (UTC)