Lesson 12a -Undetermined Coefficients

Lesson 12a -Undetermined Coefficients - NonHomogenousODE...

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Non Homogenous ODE W ill b l ki t h ODES f th f We will be looking at non homogenous ODES of the form: ) ( ' ' ' x g cy by ay We know that we can solve: Let y h be the general solution to the homogenous system above, and let y p be the 0 ' ' ' cy by ay particular solution to . The our final general solution will be y = y h +y p to the equation ) ( ' ' ' x g cy by ay ) ( ' ' ' x g cy by ay Why? Thi if fi d j t ) ( ' ' ' x g cy by ay This means if we can find just one particular solution that makes it = g(x), we will have our general solution based on solving the linear ) ( )' ( ' )' ( p h p h p h y y c y y b y y a ) ( ) ' ' ( ) ' ' ' ' ( p h p h p h y y c y y b y y a homogonous systems we did before! p p p h h h y by ay cy by ay ' ' ' ' ' ' ) ( 0 x g
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Undetermined Coefficients Th id f d t i d ffi i t i t ith bl t The idea of undetermined coefficients is to come up with a reasonable guess as to what could be substituted as y and make it equal to g(x) on the other side. But we do not know what the coefficient will be. Instead we write the coefficients as constants then substitute them into the equation to make sure left side = as constants, then substitute them into the equation to make sure left side = right side. B t h t ld b bl l ti ? But what would be a reasonable solution? We should never choose a candidate for our particular solution to be something in our homogenous solution. (All solutions to the homogenous produce 0, so how could it produce g(x)?)
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