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
6) ydx-xdy=*Q. V. (9) y dx~xdy*=*xy dx. I. x -4- cos x) dy + (cos x sin x) ' dx = 0. in * tan xdy=* cot y dx. dyQ. DIFFERENTIAL EQUATIONS 14 A homogeneous equation of the Homogeneous equations. and order degree is one which can be written in the form 15. first dx To test whether a function of x and y can be written of the right-hand side, y - x =v or form vx. e. the x's all cancel, the test is satisfied. _, ,.. Jbx. (i). dii jax 2 = -o; + 3 2 dy = dx . / becomes zx 1 4- v2 - _. This equation .
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.
14. Variables separate. dx Ex. (i). In the equation x = tan y . dy, the left-hand side involves x only and the right-hand side y only, so the variables are separate. e. cos y log (xcos t/)=c, BCOS t/=ec ~a, Ex. + c, Tx^ Xy (ii). say. ' The variables are not separate at present, but they can so. Multiply by dx and divide by y. We get easily be made ~-%xdx. y log y Integrating, As = x 2 + c. we may put c is arbitrary, it equal to log a, where a is another arbitrary constant. Thus, (1) (2) (3) y^ae^.