A Treatise On Differential Equations
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Given x'^, = {x''{-2xy)^-iy\ This equation proves homogeneous on assuming x to be of the degree 1, y of the degree 2, -^ of the degree 1, and (I'll -— of the degree 0.
Changing then, according to the" formula (24), x into 1, y into z, -f^ into w + 2z, and -r^ into ^^ ^- + S^t + 2^, we have •^ dx dx dz u^ + Su + 2z= [1 + 2z) {u + 2z) -- 4iz' (a), 220 HOMOGENEOUS EQUATIONS. [CH. X.
wliicli is reducible to .g-H2-2.)=0.
This is resolvable into two equations, viz.
dz The first gives on integration u...^[z-Vf±c\ Hence, since -Tn = '^} ^^ have nil ;^ + 2-2^ = 0, u = Q (6).
{z-rf±c^' Hence, replacing 6 by log a?, and ^ by — 2 , we have log oj = - tan"' ^^^ — 2- + c', or - log ^ — )- f^ + c', ° c cic^ 2c ^ ^ — (1 — c) a;^ ' the rational forms of the integral required, dz d9 dz The factor w = in (6) giving ^ = 0, or ^ = c, leads to the singular solution y = ca?, ^ Class III. Equations which are homogeneous with re- spect to ^,^, ^,&c.
Properly speaking, this class constitutes a limit to the class just considered.
What to read after A Treatise On Differential Equations?
You can find similar books in the "Read Also" column, or choose other free books by Boole, George, 1815-1864 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Given x'^, = {x''{-2xy)^-iy\ This equation proves homogeneous on assuming x to be of the degree 1, y of the degree 2, -^ of the degree 1, and (I'll -— of the degree 0.
Changing then, according to the" formula (24), x into 1, y into z, -f^ into w + 2z, and -r^ into ^^ ^- + S^t + 2^, we have •^ dx dx dz u^ + Su + 2z= [1 + 2z) {u + 2z) -- 4iz' (a), 220 HOMOGENEOUS EQUATIONS. [CH. X.
wliicli is reducible to .g-H2-2.)=0.
This is resolvable into two equations, viz.
dz The first gives on integration u...^[z-Vf±c\ Hence, since -Tn = '^} ^^ have nil ;^ + 2-2^ = 0, u = Q (6).
{z-rf±c^' Hence, replacing 6 by log a?, and ^ by — 2 , we have log oj = - tan"' ^^^ — 2- + c', or - log ^ — )- f^ + c', ° c cic^ 2c ^ ^ — (1 — c) a;^ ' the rational forms of the integral required, dz d9 dz The factor w = in (6) giving ^ = 0, or ^ = c, leads to the singular solution y = ca?, ^ Class III. Equations which are homogeneous with re- spect to ^,^, ^,&c.
Properly speaking, this class constitutes a limit to the class just considered.
What to read after A Treatise On Differential Equations?
You can find similar books in the "Read Also" column, or choose other free books by Boole, George, 1815-1864 to read onlineMoreLess
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