By H. T. H. Piaggio
A arithmetic textbook on differential equations.
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Extra info for An Elementary Treatise on Differential Equations and Their Applications
From Partington's Higher Mathematics for Students of p. ] CHAPTER III LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 23. The equations to be discussed in this chapter are of the dn ~! y form dy where f(x) is a function of x, but the p's are all constant. These equations are most important in the study of vibrations This will be of all kinds, mechanical, acoustical, and electrical. by the miscellaneous examples at the end of the chapter. * We shall also discuss systems of simultaneous equations of this form, and equations reducible to this form by a simple transformation.
2. *Thua excluding a function 0=0. like y/x, which it* indeterminate when x0 and GRAPHICAL REPRESENTATION 7 curve of minima. At the point of intersection the tangent is parallel to Ox. After this we ascend again, so meeting the curve of inflexions. After crossing this the characteristic becomes convex upwards. It still ascends. Now the figure shows that if it cut the curve of minima again y Fro. 2. the tangent could not be parallel to Ox, so becomes asymptotic to it. it cannot cut The other characteristics are of similar nature.
Expand ynj^ in a series of D. Hence, adding the complementary function, the solution suggested for y = \ (x* is Ex. - J) +A cos 2>x + B sin 2x. (ii). by partial fraction8j Adding the complementary function, the solution suggested Is **> -)- -96 96 Hence the solution of D 2 (Z) a + 4) y = 96x 2 y - 2x* 6x* + 4 Alternative method. 'iVi2~T coa 2x + should be B sin 2x + E + Fx. for DIFFERENTIAL EQUATIONS 38 This gives an extra term complementary function. * The method adopted 3, which however, included in the is, and where F(D) does not Suppose the expansions have been obtained by ordinary long division.