Question

4. [15 pts.] For 2y' = -tan(t)(y2 – 1) (a) find the general solution (solve for y(t)); (b) solve an initial value problem y(0) = -1/3 (state the domain of definition of the solution).

Answer 1

We solve the given differential equation by using seperation of variables method.
Given the differential equation is 2y'= - tan [(y?-1) 7204 = -tan (1) (y2-1) ► 2dy 2011 -tan(A) dt Integrating both side + Toevoegen L-tortyzat Constant 2 ln/y- lor (Cost) tlhe [i yt Where is the arbitrany 7 In = do[ccost) (y) C Cost y1 = Cytige cost y-t=clyticost >y-1= cycost + ccost y(ccost-1) = -1-cCost →ly (t) = 1+ c cost I-ccost which is the General solution of the given differential equation put, t=0 in y(o)= [ Given that, yo)=-43 I+C T-C

te v -1/3 -Ite = 3+ 3C 20=-4 C = -2 From 1-2Cost y(t) H2005+ which is the unique Solution 96 the given differential equation with initial Value Problem. y (t) exists when 1 +2 Cost to cost & cost & Cob211 + + €2n11 I 21 261. .. Domain of definition of the solution = {R-(200*3)}