Order of a Differential Equation
Definition
The order of a differential equation is the order of the highest-order derivative present in the equation.
Examples — Finding Order
| Differential Equation | Highest Derivative | Order |
|---|---|---|
| $\dfrac{dy}{dx} = x^2 + 1$ | $\dfrac{dy}{dx}$ | 1 |
| $\dfrac{d^2y}{dx^2} + y = 0$ | $\dfrac{d^2y}{dx^2}$ | 2 |
| $\dfrac{d^3y}{dx^3} - 6\dfrac{dy}{dx} = 0$ | $\dfrac{d^3y}{dx^3}$ | 3 |
| $y'' + 3y' + 2y = e^x$ | $y''$ | 2 |
Degree of a Differential Equation
Definition
The degree of a differential equation is the power (exponent) of the highest-order derivative, after the equation is made free from radicals and fractions in its derivatives.
Important: Degree is defined only when the DE is a polynomial in its derivatives. If it involves $\sin(y')$, $e^{y''}$, etc., the degree is not defined.
Examples — Finding Degree
| Differential Equation | Order | Degree |
|---|---|---|
| $\left(\dfrac{dy}{dx}\right)^2 + y = 0$ | 1 | 2 |
| $\left(\dfrac{d^2y}{dx^2}\right)^3 + x = 0$ | 2 | 3 |
| $\dfrac{d^2y}{dx^2} + \sin\!\left(\dfrac{dy}{dx}\right) = 0$ | 2 | Not defined |
| $\sqrt{1 + \left(\dfrac{dy}{dx}\right)^2} = \dfrac{d^2y}{dx^2}$ | 2 | 2 (after squaring) |
Worked Example — Clearing Radicals
Find the order and degree of: $\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \frac{d^2y}{dx^2}$
Step 1: Square both sides to remove the radical:
$$1 + \left(\frac{dy}{dx}\right)^2 = \left(\frac{d^2y}{dx^2}\right)^2$$
Step 2: Highest order derivative is $\dfrac{d^2y}{dx^2}$ → Order = 2
Step 3: Power of $\dfrac{d^2y}{dx^2}$ is $2$ → Degree = 2
Linear vs Non-Linear DEs
Linear DE
A differential equation is linear if the dependent variable $y$ and all its derivatives appear with degree 1 only, and are not multiplied together.
| Equation | Linear? | Reason |
|---|---|---|
| $\frac{dy}{dx} + y = x$ | ✅ Yes | $y$ and $y'$ both have power 1 |
| $\frac{d^2y}{dx^2} + 3y = 0$ | ✅ Yes | All derivatives appear linearly |
| $y\frac{dy}{dx} = x$ | ❌ No | $y$ and $y'$ are multiplied |
| $\left(\frac{dy}{dx}\right)^2 + y = 0$ | ❌ No | $y'$ has power 2 |
🔑 Key Takeaways
- Order = order of the highest derivative present.
- Degree = power of the highest derivative (after clearing radicals/fractions).
- Degree is undefined if the DE is not a polynomial in its derivatives.
- Order and degree are always positive integers.
- A DE is linear if $y$ and all derivatives appear with power 1 only.