Question

Use separation of variables with i =-16 to find a product solution to the following partial differential equation, си y + 0 ar2 ay that also satisfies the conditions u(0,0) = 6 and ux(0,0) = 3.

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Answer 1

Given partial differential eq.n is Y 2²4 2x² + 24 o ay With Conditions uco,ol=6 and Ux(0,0) = 3 Applying the method of Seperation of variables by taking ucxiy) Xox) Ycy) 3 the eqn ♡ reduces to x" y y y" - d² (say) (separation Constant' X given d= -16 therefore x"+d²x te X" + 256X = 0 4 from eq. (2 X(0) = 6, X'(0) = 3 0 + 116 on solving eqh G A.E. @q. D?+ 256 = 0 *(x- So, X (x) a Sin 16x + C Co8168 using eqon © Xcol=6 [C2 = 6 and X'(xl 160, 6816x - 16C2 Sin 16x Now, X 'co) = 3 =) 16C = 3 -> G 3/6

putting the value of G and Ca in in equ ® We obtain X(x) 3 Sin 16x + 6 Co816x 16 The 2nd equ of Set (a) now becomes 2 dy dy dy Y which is Linear diff equr therefore -fod y dy I, F. - dy e hot d=-16 C given) Soi 2564 Is F Now, Y cyl. I.F. Jo. If.dy to = ) 2564 e Y(y) c 2564 or У (9) ce 8 from eq.n 2 Yol = 6 C 3560 So, е е Tc=6 Hence, 2564 У (9) 6 e (9 Now, from @ # and 9 We have ucx,y) = 2564 6 e ( Sin 16x + 6 60816x)