Q.Determine the order and degree, if defined, of the differential equation: $y' + y = e^x$

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. …

Why Order and Degree Matter

When we classify a differential equation, we're asking two simple questions:

1. Order: What is the highest derivative that appears in the equation?
2. Degree: If we write the equation as a polynomial in the derivatives, what power is that highest derivative raised to?

These two numbers tell us a lot about the equation's behaviour and what methods we can use to solve it. For the equation $y' + y = e^x$, both answers are straightforward — but let's walk through the reasoning carefully.

Step-by-Step Solution

1. Identify the highest derivative

Look at each term in $y' + y = e^x$:

• $y'$ is the first derivative of $y$ with respect to $x$.
• $y$ is the function itself (zeroth derivative).
• $e^x$ is a known function of $x$, not a derivative.

The highest derivative present is $y'$ — there is no $y''$, $y'''$, or higher. So the order is 1.

2. Check if degree is defined

Degree is defined only when the equation can be written as a polynomial in the derivatives — meaning no fractional powers, no trigonometric functions of derivatives, no absolute values, etc. Our equation is:

$y' + y = e^x$ …

Method: Order and degree with a forcing term on the right

Use this on equations like $y'+y=e^x$, where a function of $x$ sits on the right.

Steps

Step 1: Order = highest derivative present.

Only $y'$ appears, so the order is $1$.

Step 2: The right-side $e^x$ is a function of $x$, not a derivative.

A forcing term like $e^x$ never changes the order or the polynomial-in-derivatives status. …

Common Mistakes

Mistake 1: Thinking the $e^x$ term changes the classification.

Why it's wrong: in $y'+y=e^x$ the $e^x$ is a function of the independent variable $x$ on the right; it affects neither the order nor the degree. Correct approach: order $=1$, degree $=1$. …