1. Idea of Forming a Differential Equation
The aim of forming a differential equation is to start from a family of functions that contain one or more arbitrary constants and eliminate those constants using differentiation. This produces a relation involving the variable, the dependent function, and its derivatives — a differential equation.
In simple terms: start with a general family of curves → differentiate → remove arbitrary constants → the resulting relation is the required differential equation.
1.1. Why elimination is needed
Arbitrary constants represent the flexibility of a family of solutions. A differential equation captures the essential relationship between a function and its derivatives, leaving no arbitrary constants behind. Removing these constants ensures the equation describes the entire family without referencing any specific member.
2. Definition
Formation of a differential equation means finding a differential equation whose solution is a given family of curves. This is done by differentiating the family enough times to get as many equations as there are arbitrary constants, and then eliminating those constants.
If the family contains n arbitrary constants, the formed differential equation is usually of order n.
2.1. Key idea
Number of arbitrary constants → number of differentiations needed → order of the resulting differential equation.
3. General Procedure for Formation
The process of forming a differential equation is systematic. The steps are:
3.1. Step-by-step method
- Start with the given family of functions containing arbitrary constants, such as:
\(y = A e^x + B e^{-x}.\)
- Differentiating the expression as many times as needed (as many constants are present).
- Write all differentiated forms clearly.
- Eliminate the constants by solving the system of equations formed.
- The final relation involving only \(x\), \(y\), and derivatives of \(y\) is the required differential equation.
4. Forming Differential Equations by Eliminating One Constant
When the family contains a single arbitrary constant, only one differentiation is needed. After differentiating, eliminate the constant by expressing it from the original equation and substituting it into the derivative.
4.1. Example
Given the family:
\(y = A e^x.\)
Differentiating:
\(\frac{dy}{dx} = A e^x.\)
From the original equation, \(A e^x = y\). Substitute this into the derivative:
\(\frac{dy}{dx} = y.\)
Thus the formed differential equation is:
\(\frac{dy}{dx} - y = 0.\)
5. Forming Differential Equations by Eliminating Two Constants
When the family contains two arbitrary constants, differentiate twice. Then eliminate both constants from the equations obtained from differentiation.
5.1. Example
Given the family:
\(y = A \cos x + B \sin x.\)
Differentiating once:
\(\frac{dy}{dx} = -A \sin x + B \cos x.\)
Differentiating twice:
\(\frac{d^2y}{dx^2} = -A \cos x - B \sin x.\)
Notice that:
\(\frac{d^2y}{dx^2} = -y.\)
This eliminates both \(A\) and \(B\). Thus the formed differential equation is:
\(\frac{d^2y}{dx^2} + y = 0.\)
6. Forming Differential Equations of Higher Order
For families with more than two constants, continue differentiating until the number of derived equations matches the number of constants. The procedure remains the same: differentiate → eliminate constants → obtain the differential equation.
6.1. Note on practical elimination
Sometimes constants can be eliminated elegantly using algebraic manipulation or by recognizing patterns. In other cases, systematic substitution is needed. The final equation should contain no arbitrary constants.
7. Worked Examples
The following examples illustrate the formation process clearly.
7.1. Example 1: Curve with one constant
Given the family:
\(y = (A + x)^2.\)
Differentiating:
\(\frac{dy}{dx} = 2(A + x).\)
From the original equation:
\(A + x = \sqrt{y}.\)
Substitute into the derivative:
\(\frac{dy}{dx} = 2\sqrt{y}.\)
The formed differential equation is:
\(\frac{dy}{dx} - 2\sqrt{y} = 0.\)
7.2. Example 2: Curve with two constants
Given the family:
\(y = A e^{kx}.\)
This expression contains two constants: \(A\) and \(k\). Differentiate twice:
\(\frac{dy}{dx} = A k e^{kx},\)
\(\frac{d^2y}{dx^2} = A k^2 e^{kx}.\)
Eliminate \(A\) and \(k\) using ratios:
\(\frac{d^2y}{dx^2} \, y = \left( \frac{dy}{dx} \right)^2.\)
This is the required differential equation.
8. Notes and Observations
Important points to remember about formation:
- The number of arbitrary constants determines the order of the differential equation formed.
- Differentiate as many times as the number of constants.
- The final relation must not contain any arbitrary constants.
- The process may involve substitution, algebraic rearrangement, or pattern recognition.