When degree is not defined
If a derivative appears inside sin, cos, e, log (e.g. \(y''+e^{y'}=0\)), the equation is not a polynomial in derivatives and its degree is not defined.
Differential Equations
Order and Degree
Understand
The order of a differential equation is the order of the highest derivative in it. The degree is the highest power of that highest-order derivative, provided the equation is a polynomial in all its derivatives.
| Equation | Order | Degree |
|---|---|---|
| \(\frac{dy}{dx}+y=e^x\) | 1 | 1 |
| \(\left(\frac{d^2y}{dx^2}\right)^3+\left(\frac{dy}{dx}\right)^4=0\) | 2 | 3 |
| \(y''+\sin(y')=0\) | 2 | not defined |
| \(\sqrt{1+(y')^2}=y''\) | 2 | 2 (after squaring) |
Order and degree are always positive integers (when the degree is defined). Remove radicals and fractional powers of derivatives before reading the degree. If a derivative appears inside \(\sin\), \(e^{(\cdot)}\), \(\log\) etc., the degree is not defined.
Key Concepts
- Order: highest derivative.
- Degree: power of the highest derivative after making the equation polynomial in derivatives.
- Transcendental functions of derivatives ⇒ degree not defined.
Key Points
Common Mistakes
Degree with radicals
For \(\left(1+(y')^2\right)^{3/2}=y''\), square both sides first: the degree is 2, not \(\frac23\) or undefined.
Practice & Topic Test
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Find the highest derivative (order); clear radicals and read its power (degree).
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