Q.Determine the order and degree, if defined, of the differential equation: $y' + 5y = 0$

The key idea is that order is the highest derivative present, and degree is the power of that highest derivative when the equation is polynomial in derivatives.

Step 1: Identify the highest derivative. Here $y' = \frac{dy}{dx}$ is the only derivative, so the order is 1.

Step 2: The equation is already polynomial in $y'$ (no fractional powers, no transcendental functions of $y'$). The highest derivative $y'$ appears to the first power.

Step 3: Therefore, the degree is 1.

Why this approach works

When we talk about the order of a differential equation, we mean the highest derivative that appears in the equation. For degree, we mean the power of that highest derivative — but only after the equation is written as a polynomial in derivatives (no radicals, no fractions inside derivatives). Here, the equation is already clean: $y' + 5y = 0$. There is only one derivative, $y'$, and it is not inside a square root, a fraction, or any other function. So both order and degree are immediately clear.

Step-by-step solution

1. Identify the highest derivative present.

   The equation is $y' + 5y = 0$. The only derivative is $y'$ (which is $\frac{dy}{dx}$). There is no $y''$, $y'''$, or any higher derivative. So the order is $1$.

2. Check if the equation is polynomial in the highest derivative.

   The term $y'$ appears as itself, not inside a sine, exponential, square root, or denominator. The equation is already a polynomial in $y'$ (specifically, $1 \cdot y' + 5y = 0$). So the degree is defined.

3. Find the power of the highest derivative.

   The highest derivative $y'$ is raised to the power $1$ (since it is simply $y'$, not $(y')^2$ or $\sqrt{y'}$). Therefore, the degree is $1$.

4. State the result. Order = 1, Degree = 1.

Method: Order and degree of a first-order equation

Use this whenever a question asks only to classify an equation like $y'+5y=0$.

Steps

Step 1: Order = highest derivative present.

The only derivative is $y'$, so the order is $1$.

Step 2: Check polynomial form, then read the degree.

$y'$ is not inside a root or a transcendental function, so the degree is defined and equals the power of the highest derivative — here $y'$ to the first power, so degree $1$.

Step 3: Ignore the dependent variable's power.

The $y$-term (and any constant coefficient) has no bearing on either order or degree. Both are read entirely off the derivatives.

Common Mistakes

Mistake 1: Judging the degree from the power of $y$.

Why it's wrong: in $y'+5y=0$ the dependent variable $y$ and its coefficient $5$ are irrelevant to the degree. Degree is the power of the highest derivative, $y'$, which is $1$. Correct approach: order $1$, degree $1$, read entirely off $y'$.

Mistake 2: Mixing up order and degree.

Why it's wrong: reporting them in the wrong slots. Correct approach: order = highest derivative (here $y'$, so $1$); degree = its power ($1$).