Q.Determine the order and degree, if defined, of the differential equation: $y''' + 2y'' + y' = 0$

The key idea is that order is the highest derivative present, and degree is the power of that highest derivative after the equation is made polynomial in derivatives (no radicals or fractions in the derivatives).

Step 1: Identify the highest derivative. Here, $y'''$ is the third derivative, so the order is 3. …

Why this approach works

The order of a differential equation is simply the highest derivative that appears. No deeper trick — just look for the most number of primes (or the largest $n$ in $y^{(n)}$). The degree is trickier: it is the power of the highest-order derivative after the equation has been cleared of radicals, fractions, and any non-polynomial expressions in the derivatives. Here the equation is already a clean polynomial in $y'''$, $y''$, and $y'$, so the degree is just the exponent on $y'''$.

A common confusion: students think degree means "the highest power of $y$" or "the highest power of any derivative." It is specifically the power of the highest-order derivative only.

Step-by-step solution

1. Identify the highest derivative. The given equation is

The derivatives present are $y'$ (first), $y''$ (second), and $y'''$ (third). The highest is $y'''$, so the order is $3$.

2. Check if the equation is polynomial in derivatives.

   The equation contains only integer powers of $y'''$, $y''$, and $y'$ — each appears to the first power. There are no square roots, no fractions like $\frac{1}{y''}$, no trigonometric or exponential functions of derivatives. So the degree is defined.

3. Read off the degree. …

Method: Order and degree of a linear constant-form equation

Use this on equations like $y'''+2y''+y'=0$, where every derivative appears once.

Steps

Step 1: Order = highest derivative present.

$y'''$ gives order $3$.

Step 2: Check the polynomial condition.

All terms are first powers of derivatives with no radicals or transcendental functions, so the degree is defined. …

Common Mistakes

Mistake 1: Setting the degree equal to the order.

Why it's wrong: for $y'''+2y''+y'=0$ the order is $3$, but the degree is the power of $y'''$, which is $1$ — the two need not match. Correct approach: order $=3$, degree $=1$.

Mistake 2: Being unsure whether the degree is defined. …