A solution of a differential equation is an expression to show the dependent variable in terms of the independent one(s) I order to satisfies the relation between variables. The general solution includes all possible families of solutions and it includes arbitrary constants (in the case of an ODE) or arbitrary functions (in the case of a PDE.) A solution which has no arbitrary constants or functions is called a particular solution.
If the given family P of curves depends on only one parameter then we can represent the equation of the form
P(x, y, a) = 0
For a family of parabolas, y2=ax can be represented by:
f(x, y, a): y2 = ax
Differential equation with respect to x for an equation containing y’, y, x and a can be written as:
g(x, y, y’, a) = 0
We can then write the required differential equation by removing a from the above equations:
F(x, y, y’) = 0
Now consider a family of curves, Q that depends on a and b, then:
Q (x, y, a, b) =0
Again, we differentiate the equation to obtain:
G(x, y, y’, a, b) = 0
However, elimination of two parameters is not possible, hence we need a third equation which we can obtain by using second-order differentiation:
h(x, y, y’, y’’, a, b) = 0
Likewise, we obtain the differential equation by eliminating a and b from the equation:
F(x, y, y’, y”) = 0
The order of a differential equation representing a family of curves is the same as the number of arbitrary constants that are shown in the equation of the family of curves.
Example: Consider the family of curves y = mx, where m is an arbitrary constant. Form the DE for this family
Solution:
We differentiate y = mx on both sides to obtain: $\frac{dy}{dx}=m$
We then substitute the constant value, m with differential obtained from the original equation:
$y=\frac{dy}{dx}x$
Or
$x\frac{dy}{dx}-y=0$
This equation is free from any parameter, therefore we can use it as a differential equation.
Example: Consider a family of curves: y = a sin (x + b). Form the differential equation:
We differentiate the equation on both sides to obtain DE with respect to x:
$\frac{dy}{dx}=a\cos (x+b)$
$\frac{d^{2}y}{dx^{2}}=-asin (x+b)$
The two equations above can be used to eliminate a and b from the original equation:
Hence, $\frac{d^{2}y}{dx^{2}}+y=0$
This equation is the required DE which is free from any arbitrary constants.