Question

2. a. Show that y² + x – 3 = 0 is an implicit solution to dy/dx = -1/(2y) on the interval (-0, 3). b. Show that xy3 – xy: sin x = 1 is an implicit solution to dy_(x cos x + sin x - 1) y 3(x - x sin x) on the interval (0, /2).

Answer 1

Given that - ² + x3 =0 - = y + 3 x the domain of the function -Coms differentiate equation is (-co ,3) with respect to se du 27. du + de f (a) = 0 - zy, dy + 1-0 -2 23 dy - -- I do 24 - Hence tx-3 = 0 is an implicit bolution of the diff. equation 2.4² x 3 sina = 1 2-asina 3 -0 Now, differential equation with respect to a dx - Casinx) - (y) - L - XL05 - sinx = -35 %. mulbolytiwide both sides by Y. G-acosx -sinx) = -35 de dx dra

= (1-2cosa-sinady - dy da plug y = ne-asinx. in the above equation - da osx + Sinx - 1) Y 3 x -usina) Thus xy² x y sina = 1 is an implicite solution the differential equation iff - xsina to when at x = 0 , 0-o-sino =0 at a= 2 - The sin (T) = T T = 0 The Hence at x=0, Tll the above equation is not a solution of the differential equation. But when xt to Mlad, masina x-usinxto Thus. 23- xy sina = 1 is the implicit solution of the differential equation dy = Cucose + sinx-DY da 3(n-usine 2 on the interval (0, 1/2)