HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS
15.3.5 Equations homogeneous inxoryaloneIt will be seen that the intermediate equation (15.85) in the example of the
previous subsection was simplified by the substitutionx=et, in that this led to
an equation in which the new independent variabletoccurred only in the form
d/dt; see (15.86). A closer examination of (15.85) reveals that it is dimensionally
consistent in the independent variablextaken alone; this is equivalent to giving
the dependent variable and its differential a weightm= 0. For any equation that
is homogeneous inxalone, the substitutionx=etwill lead to an equation that
does not contain the new independent variabletexcept asd/dt. Note that the
Euler equation of subsection 15.2.1 is a special, linear example of an equation
homogeneous inxalone. Similarly, if an equation is homogeneous inyalone, then
substitutingy=evleads to an equation in which the new dependent variable,v,
occurs only in the formd/dv.
Solvex^2d^2 y
dx^2+xdy
dx+
2
y^3=0.
This equation is homogeneous inxalone, and on substitutingx=etwe obtain
d^2 y
dt^2+
2
y^3=0,
which does not contain the new independent variabletexcept asd/dt. Such equations
may often be solved by the method of subsection 15.3.2, but in this case we can integrate
directly to obtain
dy
dt
=
√
2(c 1 +1/y^2 ).This equation is separable, and we find
∫
dy
√
2(c 1 +1/y^2 )
=t+c 2.By multiplying the numerator and denominator of the integrand on the LHS byy, we find
the solution √
c 1 y^2 +1
√
2 c 1=t+c 2.Remembering thatt=lnx, we finally obtain
√
c 1 y^2 +1
√
2 c 1
=lnx+c 2 .Solution method. If the weight ofxtaken alone is the same in every term in the
ODE then the substitutionx=etleads to an equation in which the new independent
variabletis absent except in the formd/dt. If the weight ofytaken alone is the
same in every term then the substitutiony=evleads to an equation in which the
new dependent variablevis absent except in the formd/dv.