Direction fields are analytic solutions to ordinary differential equations. O True O False In most cases the unique solution of a given differential equation, hence a particular solution, is not obtained from a general solution.

General solutions for Legendre differential equations (2x + 3)2dy dx2 + (2x + 3)+y = 0 IS..... dx

Differential Equations

Problem. Substitution method is a frequently used technique in mathematics. Through an appropriate substitution, a new (and unsolvable) problem can be turned into an old (and solvable) problem. In calculus, we used this method a lot in integrals. This method can be employed in differential equations, either. The differential equation dy dx -x + √x² + y² Y describes the shape of a plane curve that will reflect all incoming light beams to the same point and could be a model for the mirror of a reflecting telescope, a satellite antenna, or a solar collector. See a previous written assignment. This equation can be solved by means of the substitution u = x² + y² with y = y(x). Solve the equation following the steps below. 1. Express du in terms of x, y and dy. (Hint: Use the chain rule.) 2. Use the differential equation of y(x) to form a differential equation of u(x).