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SeriesSolutions, variable coefficient linear differential equation Classify the points of the differential equation, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> y= </mtext> <munderover> <sum/> <mtext> n=0 </mtext> <mtext> ∞ </mtext> </munderover> <mmultiscripts> <mtext> a </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mmultiscripts> <mrow> <mtext> (x- </mtext> <mmultiscripts> <mtext> x </mtext> <mtext> 0 </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> </mrow> <none/> <mtext> n+r </mtext> </mmultiscripts> </mrow> </math> Technique to determine coefficients in series solution (recurrence relations) and r (indicial equation), <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> y= </mtext> <munderover> <sum/> <mtext> n=0 </mtext> <mtext> ∞ </mtext> </munderover> <mmultiscripts> <mtext> a </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mmultiscripts> <mrow> <mtext> (x- </mtext> <mmultiscripts> <mtext> x </mtext> <mtext> 0 </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> </mrow> <none/> <mtext> n </mtext> </mmultiscripts> </mrow> </math> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> y= </mtext> <munderover> <sum/> <mtext> n=0 </mtext> <mtext> ∞ </mtext> </munderover> <mmultiscripts> <mtext> a </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mmultiscripts> <mrow> <mtext> (x- </mtext> <mmultiscripts> <mtext> x </mtext> <mtext> 0 </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> </mrow> <none/> <mtext> n+r </mtext> </mmultiscripts> </mrow> </math>, Classify the points of the differential equation Irregular Singular Point, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mtext> y= </mtext> <mmultiscripts> <mtext> x </mtext> <none/> <mtext> r </mtext> </mmultiscripts> </math> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> y= </mtext> <munderover> <sum/> <mtext> n=0 </mtext> <mtext> ∞ </mtext> </munderover> <mmultiscripts> <mtext> a </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mmultiscripts> <mrow> <mtext> (x- </mtext> <mmultiscripts> <mtext> x </mtext> <mtext> 0 </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> </mrow> <none/> <mtext> n+r </mtext> </mmultiscripts> </mrow> </math>, Irregular Singular Point Grad school techniques: asymptotics, dominant balance, Not an Euler Equation: solution is not a Taylor series assume solution is <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> y= </mtext> <munderover> <sum/> <mtext> n=0 </mtext> <mtext> ∞ </mtext> </munderover> <mmultiscripts> <mtext> a </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mmultiscripts> <mrow> <mtext> (x- </mtext> <mmultiscripts> <mtext> x </mtext> <mtext> 0 </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> </mrow> <none/> <mtext> n+r </mtext> </mmultiscripts> </mrow> </math>, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> DE has two linearly independent Taylor series solution </mtext> </mrow> </math> assume solution is <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> y= </mtext> <munderover> <sum/> <mtext> n=0 </mtext> <mtext> ∞ </mtext> </munderover> <mmultiscripts> <mtext> a </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mmultiscripts> <mrow> <mtext> (x- </mtext> <mmultiscripts> <mtext> x </mtext> <mtext> 0 </mtext> <none/> </mmultiscripts> <mtext> ) </mtext> </mrow> <none/> <mtext> n </mtext> </mmultiscripts> </mrow> </math>, Classify the points of the differential equation Ordinary Point, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> Special Case: Euler Equation </mtext> </mrow> </math> <math xmlns="http://www.w3.org/1998/Math/MathML"> <mmultiscripts> <mtext> x </mtext> <none/> <mtext> 2 </mtext> </mmultiscripts> <mmultiscripts> <mtext> y </mtext> <none/> <mtext> // </mtext> </mmultiscripts> <mtext> +αx </mtext> <mmultiscripts> <mtext> y </mtext> <none/> <mtext> / </mtext> </mmultiscripts> <mtext> +βy=0 </mtext> </math>
