User:Egm4313.s12.team11.Suh/R2.5

Problem Statement
 Problem 2.5  Find an Ordinary Differential Equation. Use $$y''+ay'+by=0\!$$ for the given basis.

ODE Solutions
Two Real Roots: $$y(x)=c_{1}e^{\lambda_{1}x}+c_{2}e^{\lambda_{2}x}\!$$ Double Real Roots: $$y(x)=c_{1}e^{\lambda_{1}x}+c_{2}xe^{\lambda_{2}x}\!$$

Reverse Engineering
$$e^{\lambda x}(\lambda^2+a\lambda+b)=0\!$$ Case 1: $$a^2-4b>0\!$$ $$\lambda^2-(\lambda_{1}+\lambda_{2})\lambda+\lambda_{1}\lambda_{2}\!$$ $$y''-(\lambda_{1}+\lambda_{2})y'+\lambda_{1}\lambda_{2}y=0\!$$ Case 2: $$a^2-4b=0\!$$ $$\lambda^2-(\lambda_{1}+\lambda_{2})\lambda+\lambda_{1}\lambda_{2}\!$$ $$y''-(\lambda_{1}+\lambda_{2})y'+\lambda_{1}\lambda_{2}y=0\!$$

R2.5 K2011 p.59 pbs.16
$$e^{2.6x}, e^{-4.3x}\!$$ $$(\lambda-2.6)(\lambda+4.3)\!$$ $$\lambda^2+1.7\lambda-11.18\!$$

ODE Form: $$y''+1.7y'-11.18=0\!$$

R2.5 K2011 p.59 pbs.17
$$e^{-\sqrt{5}x}, xe^{-\sqrt{5}x}\!$$ $$(\lambda+\sqrt{5})^2\!$$ $$\lambda^2+2\sqrt{5}\lambda+5\!$$

ODE Form: $$y''+2\sqrt{5}y'+5y=0\!$$