Differential Equations Lecture Work Solutions 190

Differential Equations Lecture Work Solutions 190 - 1e uxx...

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1e. u xx + y 2 u yy = y A =1 B =0 C = y 2 4 = 4 y 2 < 0 elliptic if y 6 For y = 0 the equation is parabolic and it is in canonical form u xx dy dx = ± 4 y 2 2 = ± iy dy y = ± idx ln y = ± ix + c ξ =ln y + ix η y ix α x α y = 1 y β = x β y u x = u β β x + u α α x = u β u y = u α 1 y + u β β y = 1 y u α u xx =( u β ) x = u ββ u = 1 y ! y u α + 1 y ( u α ) y = 1 y 2 u α + 1 y 2 u αα u + y 2 1 y 2 u α + 1
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