Formation of Differential Equations: Explained with Examples

Formation of a differential equation means working backwards: starting from a family of curves with arbitrary constants and eliminating those constants to obtain the differential equation that every member of the family satisfies. It is the reverse of solving.

This matters because the differential equations of physics are born this way. The family of all projectile paths, the family of circles with arbitrary centres, or the family of exponential growth curves each hides one differential equation, and forming it reveals the law shared by the whole family.

Key Concepts

• Arbitrary constants: the parameters, usually written c₁, c₂, …, that pick one particular curve out of the family. A family with n independent constants yields an n-th order differential equation.
• Elimination by differentiation: each differentiation produces a new relation; with n constants you differentiate n times, then use the n + 1 equations (the original plus n derivatives) to eliminate all constants.
• Constants must vanish: the final differential equation contains only x, y, and derivatives, never the arbitrary constants.
• Order matches constant count: eliminating n independent constants always raises the order to n, which is why the general solution of an n-th order equation carries n constants.

How It Works

1. Count the arbitrary constants n in the given family; the result will be an n-th order equation.
2. Differentiate n times with respect to x, producing n new equations alongside the original.
3. Eliminate the constants algebraically between these n + 1 equations.
4. Write the resulting relation between x, y, and the derivatives; that is the required differential equation.

Worked Example

Form the differential equation of the family y = c₁eˣ + c₂e⁻ˣ.

1. Count constants: there are two, c₁ and c₂, so differentiate twice; the result will be second order.
2. First derivative: dy/dx = c₁eˣ − c₂e⁻ˣ, since the derivative of e⁻ˣ is −e⁻ˣ.
3. Second derivative: d²y/dx² = c₁eˣ + c₂e⁻ˣ, because differentiating −c₂e⁻ˣ gives +c₂e⁻ˣ.
4. Eliminate: the right-hand side of the second derivative is exactly the original expression for y. Hence d²y/dx² = y, or d²y/dx² − y = 0, with no constants remaining. This second-order equation is satisfied by every curve of the family.

Common Misconceptions

• Differentiating only once for two constants. One differentiation cannot remove two independent constants; the leftover constant means the job is unfinished.
• Treating fixed numbers as constants to eliminate. Only the arbitrary parameters c₁, c₂, … are eliminated; fixed coefficients like the 2 in e^(2x) stay.
• Assuming order always equals the constant count. This holds when the constants are independent. If one constant can be absorbed into another, the effective count is smaller.

Real-World Applications

• Physics: the differential equations of motion are formed by eliminating the arbitrary launch parameters from families of trajectories.
• Engineering: beam theory forms a fourth-order equation from the family of possible deflection curves.
• Geometry: families of circles, parabolas, or orthogonal trajectories each yield a differential equation describing the whole family.
• Modelling: forming the equation from observed curve families is the first step in discovering the law behind data.

To form a differential equation, count the arbitrary constants, differentiate that many times, and eliminate every constant. The surviving relation between x, y, and derivatives is the single law obeyed by the entire family of curves.