Mixing
Particular integral of $x^2 y''-xy'+4y=cos(ln{x})+xcdot sin(ln{x})$
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I have found the complementary Function, but can't understand how to find the particular integral because with UC method. Please someone help me. Transcript of image: Q: $x^2 y'' - xy' + 4y = cos(ln x) + xsin(ln x)$ (labelled equation 1). Let $x = e^z Rightarrow z = log x$ and $xy' = Delta y$ , $x^2y'' = Delta(Delta-1)y$ where $Delta = frac{mathrm{d}}{mathrm{d}z}$ so, equation (1) becomes, $Delta(Delta - 1)y - Delta y + 4y = cos z + e^zsin z$ $(Delta^2 - 2Delta + 4) y = cos z + e^z sin z$ (labelled equation 2) Auxilliary equation of equation (2) is $m^2 - 2m + 4 = 0$ $Rightarrow m = frac{2pm sqrt{4 - 16}}{2}$ $m = 1 pm sqrt{3}i$ Therefore Complementary function, $y_c = e^z c_a cos sqrt{3}z + e^zc_2sinsqrt{3}z$ .
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