ODE types reductible to the variables separable case
There are certain of non-linear ordinary differential equations of first order (http://planetmath.org/ODE) which may by a suitable substitution be to a form where one can separate (http://planetmath.org/SeparationOfVariables) the variables.
I. So-called homogeneous differential equation
This means the equation of the form
where and are two homogeneous functions of the same degree (http://planetmath.org/HomogeneousFunction). Therefore, if the equation is written as
its right hand side is a homogeneous function of degree 0, i.e. it depends only on the ratio , and has thus the form
We can infer as well, that if one integral curve is represented by , , then also , an integral curve for any constant . Hence the integral curves are homothetic with respect to the origin; therefore some people call the equation (1) a similarity equation.
For generally solving the equation (1), make the substitution
The equation takes the form
which shows that any root (http://planetmath.org/Equation) of the equality gives a singular solution . The variables in (2) may be :
II. Equation of the form y= f(ax+by+c)
It’s a question of the equation
where , and are given constants. If is constant, then is constant, and we see that the lines constant are isoclines of the intgral curves of (3).
be a new variable. It changes the equation (3) to
Here, one can see that the real zeros of the right hand side yield lines (4) which are integral curves of (3), and thus we have singular solutions. Moreover, one can separate the variables in (5) and integrate, obtaining as a function of . Using still (4) gives also . The general solution is
Example. In the nonlinear equation
which is of the type II, one cannot separate the variables and . The substitution converts it to
where one can separate the variables. Since the right hand side has the zeros , the given equation has the singular solutions given by . Separating the variables and , one obtains
Accordingly, the given differential equation has the parametric solution
- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
|Title||ODE types reductible to the variables separable case|
|Date of creation||2013-03-22 18:06:36|
|Last modified on||2013-03-22 18:06:36|
|Last modified by||pahio (2872)|
|Defines||homogeneous differential equation|