Q.Determine the order and degree, if defined, of the differential equation: $(y''')^2 + (y'')^3 + (y')^4 + y^5 = 0$

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

Step 1: Identify the highest derivative.

Here, $y'''$ (third derivative) is present, so the order is $3$.

Step 2: Check if the equation is polynomial in derivatives. …

Why this approach works

When we talk about the order of a differential equation, we mean the highest derivative that appears. That part is straightforward — just look for the largest number of primes (or the highest derivative index). The degree, however, is trickier. It is defined only when the equation is a polynomial in the derivatives (no radicals, no fractional powers on the derivative terms). Once that condition is satisfied, the degree is the exponent of the highest-order derivative.

Here, every term is already a polynomial in the derivatives: $(y''')^2$, $(y'')^3$, $(y')^4$, and $y^5$. No square roots, no fractions like $\sqrt{y''}$ or $\frac{1}{y'}$. So the degree is well-defined and we can read it off directly.

Step-by-step solution

1. Identify the highest derivative.

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

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

   The equation is

Each term is a power of a derivative (or of $y$). There are no radicals, no fractional exponents, no trigonometric or logarithmic functions applied to any derivative. Hence the equation is a polynomial in $y'''$, $y''$, $y'$, and $y$.

3. Find the degree. …

Method: Order and degree with several powered derivatives

Use this on equations like $(y''')^2+(y'')^3+(y')^4+y^5=0$, built to blur which exponent matters.

Steps

Step 1: Order = the highest derivative present.

$y'''$ appears, so the order is $3$.

Step 2: Confirm polynomial form.

Every term is a whole-number power of a derivative — no radicals, no transcendental wrappers — so the degree is defined. …

Common Mistakes

Mistake 1: Taking the largest exponent in the equation as the degree.

Why it's wrong: in $(y''')^2+(y'')^3+(y')^4+y^5=0$ the biggest power is $4$ (or $5$), but degree is the power of the highest-order derivative $y'''$, which is $2$. Correct approach: degree $=2$.

Mistake 2: Confusing which term sets the order. …