Mastering Order and Degree of Differential Equations (Ex 9.1)

What are Differential Equations?

At its core, a differential equation is an equation that involves an unknown function and its derivatives. Think of it as a mathematical statement about how a quantity changes. For instance, if 'y' is a function of 'x', a differential equation might involve expressions like dy/dx or d²y/dx². These equations are incredibly powerful tools used across various fields like physics, engineering, economics, and biology to model dynamic systems.

For example, Newton's second law of motion, which describes how an object moves under the influence of forces, can be expressed as a differential equation. Similarly, equations describing heat flow, electrical circuits, or even the spread of a virus are all types of differential equations. They provide a precise language for describing change and are a cornerstone of advanced mathematics and its applications. Before we can solve them, we first need to understand their fundamental characteristics: order and degree.

Understanding the Order of a Differential Equation

Order of a Differential Equation

The order of a differential equation is defined as the order of the highest derivative appearing in the equation. It tells us the highest rate of change involved in the relationship. For example, dy/dx is a first-order derivative, d²y/dx² is a second-order derivative, and so on. If the highest derivative present is d³y/dx³, then the order of the differential equation is 3.

Understanding the Degree of a Differential Equation

Degree of a Differential Equation

The degree of a differential equation is defined as the highest power (exponent) of the highest order derivative, provided that the differential equation can be expressed as a polynomial in its derivatives. If the equation cannot be expressed as a polynomial in its derivatives (e.g., if a derivative is inside a trigonometric function, exponential function, or logarithm), then its degree is undefined.

Worked Examples: Finding Order and Degree

• Example 1: Find the order and degree of the differential equation: dy/dx + 5y = sin(x) Step 1: Identify all derivatives present. Here, only dy/dx is present. Step 2: Determine the highest order derivative. The highest order derivative is dy/dx, which is a first-order derivative. So, the Order = 1. Step 3: Check if the equation is a polynomial in its derivatives. Yes, dy/dx + 5y - sin(x) = 0 is a polynomial in dy/dx. Step 4: Find the power of the highest order derivative. The highest order derivative dy/dx has a power of 1. So, the Degree = 1.
• Example 2: Find the order and degree of the differential equation: (d²y/dx²)³ + (dy/dx)² + y = x Step 1: Identify all derivatives. We have d²y/dx² and dy/dx. Step 2: Determine the highest order derivative. The highest order derivative is d²y/dx² (second order). So, the Order = 2. Step 3: Check if the equation is a polynomial in its derivatives. Yes, it is. Step 4: Find the power of the highest order derivative (d²y/dx²). Its power is 3. So, the Degree = 3.
• Example 3: Find the order and degree of the differential equation: d²y/dx² + cos(dy/dx) = 0 Step 1: Identify all derivatives. We have d²y/dx² and dy/dx. Step 2: Determine the highest order derivative. The highest order derivative is d²y/dx² (second order). So, the Order = 2. Step 3: Check if the equation is a polynomial in its derivatives. Here, cos(dy/dx) involves a derivative inside a trigonometric function. This means the equation cannot be expressed as a polynomial in its derivatives. Step 4: Conclude the degree. Since the equation is not a polynomial in its derivatives, its Degree is undefined.

Exam Tips and Common Mistakes to Avoid

When determining the order and degree, students often make a few common errors. Always remember these crucial points:

1. Simplify First for Degree: If the differential equation contains radicals (like square roots) or fractions involving derivatives, clear them to express the equation as a polynomial in derivatives before finding the degree. For example, if you have √(dy/dx) + y = x, square both sides to get dy/dx + y² + x² - 2xy = 0 (after squaring (dy/dx)^{1/2} becomes dy/dx and the right side becomes (x-y)^2). Now, the degree can be found.
2. Degree Undefined Condition: The degree is undefined if any derivative appears inside a non-polynomial function like sin(dy/dx), cos(d²y/dx²), e^(dy/dx), or log(d³y/dx³). This is a frequent trick question in exams.
3. **Highest Power of Highest Order Derivative:** Don't confuse the power of a lower-order derivative with the degree. The degree is always the power of the highest order derivative, provided it's a polynomial. For instance, in (d²y/dx²) + (dy/dx)⁵ = 0, the highest order is 2, and its power is 1, so the degree is 1, not 5.
4. No Fractions in Powers: Ensure all derivatives have integer powers before determining the degree. This ties back to simplifying the equation to a polynomial form.

Practice Questions with Solutions

• Q: Find the order and degree of the differential equation: (d³y/dx³)² + x(dy/dx)⁴ + y = 0 A: Step 1: Identify all derivatives: d³y/dx³ and dy/dx. Step 2: Determine the highest order derivative: d³y/dx³ (order 3). Step 3: Check if the equation is a polynomial in its derivatives: Yes. Step 4: Find the power of the highest order derivative d³y/dx³: The power is 2. Final answer: Order = 3, Degree = 2
• Q: Determine the order and degree of: y''' + y''² + e^(y') = 0 A: Step 1: Identify all derivatives: y''' (or d³y/dx³), y'' (or d²y/dx²), y' (or dy/dx). Step 2: Determine the highest order derivative: y''' (order 3). Step 3: Check if the equation is a polynomial in its derivatives: No, because e^(y') involves a derivative inside an exponential function. Final answer: Order = 3, Degree = Undefined
• Q: Find the order and degree of the differential equation: (dy/dx)² + (d²y/dx²)⁵ = x² A: Step 1: Identify all derivatives: dy/dx and d²y/dx². Step 2: Determine the highest order derivative: d²y/dx² (order 2). Step 3: Check if the equation is a polynomial in its derivatives: Yes. Step 4: Find the power of the highest order derivative d²y/dx²: The power is 5. Final answer: Order = 2, Degree = 5
• Q: What are the order and degree of the differential equation: d²y/dx² = √(1 + (dy/dx)³)? A: Step 1: Clear the radical to express it as a polynomial in derivatives. Square both sides: (d²y/dx²)² = 1 + (dy/dx)³. Step 2: Identify all derivatives: d²y/dx² and dy/dx. Step 3: Determine the highest order derivative: d²y/dx² (order 2). Step 4: Check if the equation is a polynomial in its derivatives: Yes, after squaring. Step 5: Find the power of the highest order derivative d²y/dx²: The power is 2. Final answer: Order = 2, Degree = 2

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