example of derivative as parameter
For solving the (nonlinear) differential equation
with , according to III in the parent entry (http://planetmath.org/DerivativeAsParameterForSolvingDifferentialEquations), we differentiate both sides in regard to , getting first
Removing the denominators, we obtain
The left hand side can be factored:
Now we may use the zero rule of product; the first factor of the product in (2) yields , i.e.
whence , i.e. . Substituting this into the original equation (1) we get . Hence the general solution of (1) may be written
The second factor in (2) yields , which is substituted into (1) multiplied by :
Thus we see that , which is again set into (1), giving
Finally, we can write it
which (a variant of the so-called semicubical parabola) is the singular solution of (1).
|Title||example of derivative as parameter|
|Date of creation||2013-03-22 18:29:03|
|Last modified on||2013-03-22 18:29:03|
|Last modified by||pahio (2872)|
|Synonym||example of solving an ODE|