Thank you so much for your help!

5. Clairaut Equation Consider the Clairaut equation below (W2 3:1,\"er = 0 Traditional method like separation of variables cannot be used to obtain the solutions to this ODE. (a) (5 points) Differentiate the entire ODE with respect to :1: to show that there are 2 possible solutions resulting from y\" = 0 and 2y' a: = 0. Solve for these 2 sets of solutions y1(a:) and 1112 (3:). (b) (5 points) Plug yr and 3;; back to the ODE to determine the arbitrary constants. You should nd that one solution set is reduced to a single equation (with no arbitrary constant); this is call the singular solution. The other set is a family of lines which are the general solutions of the ODE. Sketch both yl and y2 (hand scketch only) to ShOW that the singular solution is an envelope of the general solution (tangential to all general solutions). (0) (5 points) Now considering the initial condition ($60,310) = (2, 1); apply the existence and uniqueness theorems, together with the above results, to verify that the ODE has 2 solutions passing through this point