Question

Use the method of separation of variables to find all possible product solutions of x2 Uxx + Uyy + 3 x Ux + 2uy +u = 0.

Answer 1

the-o. 2ay + Q: Use the method of separation of variables for find all possible product Solutions of x ²Uxx + Uyy + 3x Ux + A: The given Parlial differential equation (PDE) is + 3x Ux + 2lly + e=0. – 0 Let the complete solution of it u(x,y) = x (x) Y (Y). 22 Uxx + Ugy Now, U, au XY =X'Y ly- au ay azu au Uxx= = x'y ugy = xy" 2x² ay2 By putting them into o have we x² xy + xy" + 3x X'Y + 2xy't XY = 0 x(Y"+ 27') + y (x2 X" + 3x x'+ x) = 0 xx" + 3x X'+ x Y"+ 2Y' Х Y Since in L-H. S is function of 8x and R.H.S is function of y. So x²x" + 3x X't X Y"+ 2 Y' = c (Constant). Y X x²x"+ 3x x + x C - Х Y"+27 у =)

Which as Now from x² X" + 3 x x + (1-0) X = 0. a Second Order linear differential equation Its AE (Auxiliary Equation) is. x² m2 + 3x m+ (1-0)-o (xm2 2 + 2. (Xm) 3 3)?-(2) 2 + 1-c=0 + 1-c=0 (2+ - 9:1-06 (xm + ) - on 5 4 -c=0 2 xmt 3 2 5+46 4 OP, xmt 3 3/4 = 15+ 40 2 on, am 4 = om m 3 I 5+40 2 -31 15:40 2x Hence, the solution for equation ④ x = CF (Complementary Function) - 3+15+4c. -3-15440 op, X = Cie & C e - 3+15+46 - 3-5 √5 +40 op X = ce + Che 2x

Now, from Y"+ 24'+ Cy=o. which is a Second Order linear differential equation Its AE (Auxiliary Eduation) is m2 + 2m + c = 0. on, m² + 2m + 1 + (C-1) = 0. . (m2 + 1)² = 1-0 mt1 = 251-6 ( 10) m=-11 lie (170) = 1 I de-1 i (120) on, If 5a) + Cy el-Fe) >c, then solution for equation is (1+11-c) ay Y = Cze 1Lc, then solution for equation ® is y Eco Cos (de-1)y + Co Sin (de-g} If (5b 17= e-y the value of By Putting get 4 & 5a) (56) in equation ③ the complete Solution We will