Question

Newton's Method with two variables

This problem focuses on the system of simultaneous equations

$x^{x}+y^{y}=1000$

$x^{y}+y^{x}=100$

Choose any initial point $(x_{0},y_{0})$ anywhere near either solution $(x_{*},y_{*})$ of the system of equations

For this problem you need to only complete at least one iteration starting with numerical values for $(x_{0},y_{0})$ terminating with numerical values for $(x_{1},y_{1})$

Answer 1

→ Given, **+ yt = 1000 x² + y² = 100 Newton's iteration method f ,y) =o 718,4)=0 Let Crosto) be initial approximation. Then het Exoth}, Yotk) be the solution + f(xoth, Yo+k) =0 ; g (Xoth Yo+k) = 0 flroth, Yo tk) = flX0, to? + one of tk (26) = 0 = fot h(fx) + k (fy). g(xoth, Yo +k) = 9.+ h (9x). + kl 94). =0 foth (falo + k (fro = 0 { solving goth (92)ot K(fy) od har k = or where De feo frol Dx = f (fy). Dy = two Ito 9 yol I go (94) 1 (9x) 4= Xoth Yr=Yoth Leti f(x,y) = x+44-1000 g(x,y) = x'+-100 Scanned with CamScanner

ax oy fx = Of(0,1) = x*Cl+logu) fyer Cany) = y*(1+ logy) In = dg (x,y) = yoxt", ye logy og (vy)= .y*4 2 y 10g * Let Crouto) be (3,1) Then to = 3² ,-1000 = 27+1-1000 = -912 9. = 3'+,3-100 = 4-100 = -96 ffx)o = Of x^(1+ logu); 1 (fy), = 1 [1+ Ind) = 3? [1+ In 3] ). = 3.72+3* in 3 : = 56:6 = 6.29 192). 3 13'-'+ Bioni . DE 56.6 TL I 6:29 Scanned With CamScanner 356-014-| 355.01 -

Dx = -972 -96 = -6017.88 6-29 Dy = | 56.66 -972 | .-4467136 | - | na noge -(1985) = 16.95 Ks - - -/-4457.5") - 12:58 29 = Xoth = 3+16.95 = 19.95 2.4. = Yo tk = 1+12.58 = 13:58 24,4,) = (19.95,13.58) Scanned with CamScanner