Why Form a Differential Equation?
When we know the general solution (a family of curves with arbitrary constants), we can eliminate those constants by differentiating and form the corresponding differential equation.
A family of curves with $n$ arbitrary constants will yield a DE of order $n$.
Rule: If the general solution has $n$ arbitrary constants, differentiate $n$ times, then eliminate all $n$ constants to get an $n$th-order DE.
Steps to Form a DE
- Write down the given equation (family of curves) with arbitrary constants.
- Count the number of arbitrary constants — that is the order of the resulting DE.
- Differentiate the equation as many times as there are constants.
- Eliminate all arbitrary constants between the original equation and the derived equations.
- The resulting equation (without arbitrary constants) is the required DE.
Worked Examples
Example 1 — One Constant
Form a DE from $y = A\,e^{2x}$ (1 constant $A$)
Differentiate once: $\dfrac{dy}{dx} = 2A\,e^{2x}$
From original: $A = \dfrac{y}{e^{2x}}$. Substitute:
$$\frac{dy}{dx} = 2 \cdot \frac{y}{e^{2x}} \cdot e^{2x} \implies \boxed{\frac{dy}{dx} = 2y}$$
Example 2 — Two Constants
Form a DE from $y = Ae^x + Be^{-x}$ (2 constants $A$, $B$)
Differentiate once: $y' = Ae^x - Be^{-x}$
Differentiate again: $y'' = Ae^x + Be^{-x}$
Observe: $y'' = Ae^x + Be^{-x} = y$
$$\boxed{\frac{d^2y}{dx^2} - y = 0}$$
Example 3 — Circle Family
Form a DE for the family of circles $x^2 + y^2 = r^2$ (1 constant $r$)
Differentiate with respect to $x$: $2x + 2y\,\dfrac{dy}{dx} = 0$
$$\boxed{x + y\,\frac{dy}{dx} = 0}$$
This represents all circles centred at the origin.
Example 4 — Parabola Family
Form a DE for $y^2 = 4ax$ (parabolas with vertex at origin)
Differentiate: $2y\,y' = 4a \Rightarrow a = \dfrac{y\,y'}{2}$
Substitute into original: $y^2 = 4 \cdot \dfrac{y\,y'}{2} \cdot x$
$$\boxed{y = 2x\,\frac{dy}{dx}}$$
Summary Table
| Family of Curves | Constants | DE Formed |
|---|---|---|
| $y = Ae^{2x}$ | 1 | $y' = 2y$ |
| $y = Ax + B$ | 2 | $y'' = 0$ |
| $y = A\cos x + B\sin x$ | 2 | $y'' + y = 0$ |
| $y = Ae^x + Be^{-x}$ | 2 | $y'' - y = 0$ |
| $y = Ae^x + Be^{2x}$ | 2 | $y'' - 3y' + 2y = 0$ |
🔑 Key Takeaways
- Differentiate $n$ times for $n$ arbitrary constants.
- Eliminate all arbitrary constants to get the DE.
- The resulting DE has order equal to the number of constants.
- Every family of curves satisfying certain conditions corresponds to a unique DE.