Implicit Equation. Introducing the parameter.
An equation of type
where F is a continuous function, is called the first order implicit differential equation. If this equation can be solved for y', we get one or several explicit differential equations of type
that can be solved by methods covered in other sections.
The main techniques for solving an implicit differential equation is the method of introducing a parameter.
Case 1. Implicit Differential Equation of Type x=f(y,y').
Case 2. Implicit Differential Equation of Type y=f(x,y').
Case 3. Implicit Differential Equation of Type x=f(y').
Case 4. Implicit Differential Equation of Type y=f(y').
Singular Solutions.
A function φ(x) is called the singular solution of the differential equation F(x,y,y') = 0, if uniqueness of solution is violated at each point of the domain of the equation. Geometrically this means that more than one integral curve with the common tangent line passes through each point (x0,y0). Note: Sometimes the weaker definition of the singular solution is used, when the uniqueness of solution of differential equation may be violated only at some points. A singular solution of a differential equation is not described by the general integral, that is it can not be derived from the general solution for any particular value of the constant C. We illustrate this by the following example: Suppose that the following equation is required to be solved: (y')2 − 4y = 0. It is easy to see that the general solution of the equation is given by the function y = (x + C)2. Graphically, it's represented by the family of parabolas (Figure 1).
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Fig.1 |
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Besides this, the function y = 0 also satisfies the differential equation. However, this function is not contained in the general solution! Since more than one integral curve passes through each point of the straight line y = 0, the uniqueness of solution is violated on this line, and hence it is a singular solution of the differential equation.
p-discriminant
One
of the ways to find a singular solution is investigation of the
so-called p-discriminant of
the differential equation. If the function F(x,y,y') and
its partial derivatives 
and 
are
continuous in the domain of the differential equation, the singular
solution can be found from the system of equations:
Lagrange and Clairaut Equations.
Lagrange Equation
A differential equation of type
where φ(y') and ψ(y') are known functions differentiable on a certain interval, is called the Lagrange equation. By setting y' = p and differentiating with respect to x, we get the general solution of the equation in parametric form:
under condition of
where p is a parameter. Lagrange equation may also have a singular solution if the condition φ(p) − p ≠ 0 is failed. The singular solution is given by the expression:
where c is the root of the equation φ(p) − p = 0.
Clairaut Equation
The Clairaut equation has the form:
where ψ(y') is a nonlinear differentiable function. The Clairaut equation is a particular case of the Lagrange equation whenφ(y') = y'. It is solved in the same way by introducing a parameter. The general solution is given by
where C is an arbitrary constant. Similarly to the Lagrange equation, the Clairaut equation may have a singular solution that is expressed parametrically in the form:
where p is a parameter.