# An Elementary Treatise on Differential Equations and Their by H. T. H. Piaggio

By H. T. H. Piaggio

A arithmetic textbook on differential equations.

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DIFFERENTIAL EQUATIONS 6 where f(x, y) is a function of x and y having a perfectly definite value * for every pair of finite values of x and y. The curves of the family are called the characteristics of the finite equation. n dy J. /*\ EX. (l) =X (y-\\ Here Now a curve has concavity upwards when the second differential Hence the characteristics will be concave up above y = l, and concave down below this line. The maximum or minimum points lie on x = Q, since dy/dx = Q there. The characteristics near y 1, which is a member of the family, are flatter than those its coefficient is positive.

I). dii jax 2 = -o; + 3 2 dy = dx . / becomes zx 1 4- v2 - _. This equation . * is homo- geneous. Ex. 16. ~= ^-= x 2 becomes (ii). v 3 This . , is rfx not homogeneous. b Since a homogeneous equation can be solution. = putting y vx on the right-hand side, ^=f(v) by it is natural to try the effect of this substitution on the left-hand side As a matter of fact, it will be found that the equation can also. * by this substitution (see Ex. 10 of the miscelalways be solved laneous set at the end of this chapter).

E. y dy -'-r-= 1. x dx y This means that the radius vector and the tangent have gradients FIG. e. 4. that they are perpendicular. The charany radius with the origin as centre. acteristics are therefore circles of * See a paper, College of Science, " Graphical Solution," by Prof, Takeo Wada, Memoirs of th* II. No. 3, July 1917. Kyoto Imperial University, Vol. GRAPHICAL REPRESENTATION 9 In this case the singular point may be regarded as a circle of zero radius^ the limiting form of the characteristics near it, but no characteristic of finite size -, passes through it.