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
EQUATIONS OF FIRST ORDER AND FIRST DEGREE Ex. *-Z. dx y+x This * gives Putting y and proceeding as vx, dv v dx v + 1? e. x-j-** dx a + (x- y) dx - 0. (x + y)dy (ii), - 15 xi we before, , --t> = --++_. e. 2 2 Iogaj (v Substituting for z (v + 1 - tan"1 ) + log ( v 2 + 1 ) + 2 tan^t; + 2c = 0, + l)-l-2tan-1v-fa0, 2 v, log (t/ w is not (i). homogeneous form. dx homogeneous. is similar to Ex. y-x - Now y . is (ii) - of the last article, except that y-x replaced by J r , and y + x = -x= , , The l=0 and t/-fz y^Y - 3.
E. y dy This y considering y as the independent variable. e. Examples for + (1) (z (2) solution. a)|-3y=:(z x cos x~- a? + a) 6 + y(xsin 4- sc y =- 2 log + cos x)l. / + 2 -~y*(x-l). d (7) dx + x dy = e~ y sec 2 1/ dy. 22. ] . (3) -^ . sc. (4) (6) Problems. ] x 2 y - x3 ~ == y* cos x. | Orthogonal Trajectories. some geometrical problems leading to equations. T N FIG. Ex. (i). 6. Find the curve whose subtangent The subtangent TN - PN cot ^ - y ~ . is constant. We shall differential DIFFERENTIAL EQUATIONS 20 __ dx Hence -j- .
3) y e x (A cos (4) y c cosh -, (the catenary). c Find the (5) All differential x+ equation of parabolas whose axes are parallel to the axis of y. (6) All circles of radius a. (7) All circles that pass through the origin. (8) All circles (whatever their radii or positions in the plane xOy). [The result of Ex. 6 (9) Show may be used. ] that the results of eliminating a from (1) and b from are in each case y**x-~-bx* cLx x2 t 7-|-2x~ + 2z/ = ............................... (2) ...... * .....................