Non‐Homogenous ODEWill blkithODESf thfWe will be looking at non‐homogenous ODES of the form:)('''xgcybyayWe know that we can solve:Let yhbe the general solution to the homogenous system above, and let ypbe the0'''cybyayparticular solution to. The our final general solution willbe y = yh+ypto the equation)('''xgcybyay)('''xgcybyayWhy?Thiiffi d jt)('''xgcybyayThis means if we can find just oneparticular solution that makes it = g(x),we will have our general solutionbased on solving the linear)()'(')'(phphphyycyybyya)()''()''''(phphphyycyybyyahomogonous systems we did before!ppphhhybyaycybyay'''''')(0xg
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Undetermined CoefficientsThidfd tidffi it i tithbltThe idea of undetermined coefficients is to come up with a reasonable guess as towhat could be substituted as y and make it equal to g(x) on the other side. Butwe do not know what the coefficient will be. Instead we write the coefficientsas 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 th tld bbll ti?But what would be a reasonable solution?We should never choose a candidate for our particular solution to be something inour homogenous solution. (All solutions to the homogenous produce 0, so howcould it produce g(x)?)