Q.(i) The degree of the differential equation $\frac{d^2y}{dx^2}+e^{\frac{dy}{dx}}=0$ is ______.

Concept: Order and Degree of a Differential Equation — degree is defined only when the equation is a polynomial in the derivatives; transcendental terms like exponentials of derivatives make the degree undefined.

Step 1: The given equation is

$\frac{d^2y}{dx^2} + e^{\frac{dy}{dx}} = 0.$ …

The concept of degree of a differential equation is often misunderstood. Let’s clarify it first.

What does "degree" really mean?

The order of a differential equation is the highest derivative present — that’s straightforward. The degree, however, is trickier. It is defined only when the differential equation is a polynomial in all the derivatives that appear. That means every term involving $y$, $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, etc., must be a polynomial (i.e., raised to a whole-number power, no exponentials, no trigonometric functions of derivatives, no logarithms of derivatives).

If the equation contains something like $e^{dy/dx}$, $\sin(dy/dx)$, or $\log(d^2y/dx^2)$, then it is not a polynomial in the derivatives — and the degree is simply not defined.

Now let’s apply this to the given equation.

1. Identify the derivatives present The equation is:

The derivatives are $\frac{d^2y}{dx^2}$ (second order) and $\frac{dy}{dx}$ (first order). The order is clearly 2.

2. Check if the equation is a polynomial in these derivatives …

Method: Deciding when the degree is undefined

Use this to classify the degree of a DE that contains a derivative inside a transcendental function.

Steps

Step 1: Locate every derivative and how it appears

List the derivatives and check whether any sits inside $e^{(\cdot)}$, $\sin(\cdot)$, $\log(\cdot)$, etc.

Step 2: Apply the polynomial test …

Common Mistakes

Mistake 1: Assigning degree 1 because the highest derivative $\frac{d^2y}{dx^2}$ is to the first power

Why it's wrong: the term $e^{\,dy/dx}$ puts a derivative inside an exponential, so the equation is not polynomial in the derivatives. Correct approach: state the degree is not defined. …