By Brian H. Chirgwin and Charles Plumpton (Auth.)

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**Sample text**

Pn, . . with It is clear that this procedure is approximate and so gives only a rough idea of the shape of the integral curve. Also the construction breaks down if a value of xn occurs for which f(xn, y„) does not exist. This may happen if f(x, y) is discontinuous, or tends to infinity. Points where this happens are singular points. Example. The differential equation (1) has singular points at x= ± 1. The general solution of this separable differential equation is The singular points of a differential equation are important in determining the shape of the solution.

Find the differential equation of first order satisfied by the curves y2(2o-x) = xs where a is a parameter. State the differential equation of the family of orthogonal curves and show that the equation of this family is (y2 + x2)2 = C2(y2+2x2). 14. Solve the differential equation and find the equation of the family of curves any one of which is orthogonal to a solution of this differential equation. 15. (i) Find the orthogonal trajectories of the system of curves whose equation in plane polar coordinates is (ii) Prove that the system of curves whose equation is is auto-orthogonal, and sketch the curves of the system.

With It is clear that this procedure is approximate and so gives only a rough idea of the shape of the integral curve. Also the construction breaks down if a value of xn occurs for which f(xn, y„) does not exist. This may happen if f(x, y) is discontinuous, or tends to infinity. Points where this happens are singular points. Example. The differential equation (1) has singular points at x= ± 1. The general solution of this separable differential equation is The singular points of a differential equation are important in determining the shape of the solution.