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Line 28: + + + + + + + + + + + + + + − − ==Infinite Series==
may change LHS to something easier
===Second-Order Equations===
===Second-Order Equations===
==Infinite Series==
Keeping in mind how many functions can be represented with infinite sums created with Taylor series, we stop narrow-mindedly assuming a function is some combination of functions we're familiar with, and instead just assume it has an infinite series representation.
For example, let's solve <math>y''-xy'+x^2y=e^x</math> by assuming there exists <math>a_n</math> such that <math>y=\sum_{n=0}^\infty a_nx^n</math>. Then our differential equation becomes
<math>(2a_2+(3)(2)a_3x+(4)(3)a_4x^2+\dots)-x(a_1+2a_2x+3a_3x^2+\dots)+x^2(a_0+a_1x+a_2x^2+\dots) = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots</math>
Grouping like terms, we have
<math>(2a_2)+(6a_3-a_1)x+(12a_4-2a_2+a_0)x^2 + \dots </math>
so that for <math>n\geq 2</math>, the coefficient of <math>x^n</math> on the right hand side is <math>(n+1)(n+2)a_{n+2}-na_n+a_{n-2}</math>. So we need only set this equal to <math>1/(n!)</math> to find an infinite series representation of our solution.
==Functional Transformations==
==Functional Transformations==
==Non-homogeneous Equations==
==Non-homogeneous Equations==