TEAM PROJECT. Riccati Equation, Clairaut Equation. A

Chapter 1, Problem 1.5

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TEAM PROJECT. Riccati Equation, Clairaut Equation. A Riccati equation is of the form (1 I) y' + p(x)y = gp:)y 2 + hex). A Clairaut equation is of the form (12) y = xy' + g(y'). (a) Apply the transformation y = Y + lilt to the Riccati equation (1 I ), where Y is a solution of (11), and oblain for u the linear ODE u' + (2Yg - P)U = -g. Explain the effect of the transformation by writing it as y = Y + v, v = lilt. (b) Show that Y = Y = x is a solution of y' - (2x 3 + l)y = _x2y2 - X4 - X + and solve this Riccati equation. showing the details. (c) Solve y' + (3 - 2X2 sin x)y = _y2 sin x + 2x + 3x2 - X4 sin x, using (and Verifying) that y = x 2 is a solution. (d) By working "backward" from the L1-equation find further Riccati equations that have relatively simple solutions. (e) Solve the Clairautequationy = xy' + 1/y'.Hillt. Differentiate this ODE with respect to x. (f) Solve the Clairaut equation /2 - xy' + Y = 0 in Prob. 16 of 1.1. (g) Show that the Clairaut equation (12) has as solutions a family of straight lines Y = ex + gee) and a singular solution determined by g' (s) = -x, where s = y', that forms the envelope of that family.