Differential Equation

Mathematics is not about equations and numbers; it's also about studying how things change. And one of the greatest tools to use to study change is a differential equation. A differential equation is an equation that relates a function to its derivatives, revealing how one value changes with respect to another. From calculating population growth to understanding the motion of planets, differential equations are integral to science, engineering, economics, and everyday life.

 

Table of Contents

• What is a Differential Equation?
• Order of a Differential Equation
• 1st Order Differential Equation
• 2nd Order Differential Equation
• Degree of a Differential Equation
• Types of Differential Equations
• Ordinary Differential Equation (ODE)
• Partial Differential Equation (PDE)
• Linear Differential Equation
• Nonlinear Differential Equation
• Homogeneous Differential Equation
• Non-Homogeneous Differential Equation
• Exact Differential Equation
• Separable Differential Equation
• Differential Formula: Basic Rules
• Applications of Differential Equations
• Solved Problems on Differential Equations
• Conclusion
• FAQs On Differential Equation

 

What is a differential equation?

It is an equation that shows how a dependent variable changes with respect to one or more independent variables through their derivatives.

Definition

A differential equation is a mathematical equation that involves one or more functions and their derivatives. It relates a function to its rate of change.

General form:

 F(x, y, y′, y″,…, y⁽ⁿ⁾) = 0

 

Order of a Differential Equation

The order of a differential equation is defined as the highest order derivative (i.e., the derivative of the highest degree) present in the equation.

In simpler words, it tells us how many times a function has been differentiated in the equation.

 

1st Order Differential Equation

A 1st order differential equation is one that involves only the first derivative of the dependent variable with respect to the independent variable.

Form:

dy/dx + P(x)y = Q(x)

Example:

 dy/dx + y = eˣ

• Here, the highest derivative is dy/dx (1st derivative).

• So, order = 1

Explanation:
This type of equation describes how a quantity changes with respect to another (like rate of growth, speed, decay, etc.). It is widely used in exponential growth/decay, simple electrical circuits, and more.

 

2nd Order Differential Equation

A 2nd order differential equation is one that involves the second derivative of the dependent variable.

Form:

d²y/dx² + P(x)dy/dx + Q(x)y = R(x)

Example:

 d²y/dx² + 3dy/dx + 2y = 0

• Here, the highest derivative is d²y/dx² (2nd derivative).

• So, order = 2

Explanation:
2nd-order equations are used in systems with acceleration or curvature, like motion under gravity, harmonic oscillators, or beam deflection in mechanical structures.

 

Summary Table

 

Degree of a Differential Equation

The degree of a differential equation is the power of the highest order derivative, after removing all radicals and fractions involving derivatives.

Important:

• The equation must be polynomial in derivatives.

• That means no square roots, cube roots, or fractions involving derivatives.
  
  

Steps to Find a Degree:

1. Identify the highest order derivative.

2. Ensure the equation is free of roots and fractions involving derivatives.

3. The degree is the exponent of the highest order derivative.
   
   

Examples

Example 1:

 (d²y/dx²)³ + dy/dx = 5

• Highest order derivative: d²y/dx²

• Power: 3

• No roots or fractions
  → Degree = 3
  
  

Example 2:

(dy/dx)² + y = 0

• Highest order derivative: dy/dx

• Power: 2
  → Degree = 2
  
  

Example 3 (Not defined):
√(d²y/dx²) + y = 0

• Contains the square root of a derivative
  → Degree = Not defined

 

Summary Table

 

Types of Differential Equations (with Explanation and Examples)

Differential equations are classified based on different features such as the number of variables involved, the linearity, order, and the structure of the equation.

1. Ordinary Differential Equation (ODE)

An Ordinary Differential Equation involves derivatives with respect to only one independent variable. These are the most common types used in physics, engineering, and biology.

Form:

 dy/dx = f(x, y)

Example:

 dy/dx + y = eˣ

• Involves the first derivative dy/dx only.

• Only one independent variable (x).

Used in:

• Population growth

• Radioactive decay

• Newton’s laws of motion

 

2. Partial Differential Equation (PDE)

A Partial Differential Equation involves partial derivatives of a function that depends on two or more independent variables.

Form:

∂u/∂x + ∂u/∂y = 0

Example:

 ∂²u/∂t² = c² ∂²u/∂x² (Wave Equation)

• Involves partial derivatives of function u(x, t).

• More than one independent variable (x and t).

Used in:

• Heat transfer

• Sound waves

• Fluid dynamics

• Quantum mechanics
  
  

3. Linear Differential Equation

A Linear Differential Equation is one in which the dependent variable and all its derivatives appear only to the first power and are not multiplied together.

Form (1st Order):

dy/dx + P(x)y = Q(x)

Example:

 dy/dx + 2y = 4x

• y and dy/dx appear to the first power.

• No products like y·dy/dx.

Used in:

• Electric circuits

• Exponential growth/decay

• Linear motion

 

4. Nonlinear Differential Equation

A Nonlinear Differential Equation is one where the dependent variable or its derivatives appear to a power greater than one, or are multiplied together.

Example:

(dy/dx)² + y = 0

• dy/dx is raised to power 2 → non-linear.

Used in:

• Nonlinear systems

• Predator-prey models

• Chaos theory

• Chemical reactions

 

5. Homogeneous Differential Equation

A Homogeneous Differential Equation is one where each term is a function of the same degree in the dependent and independent variable.

Form:

 dy/dx = F(y/x) or dy/dx = (ax + by)/(cx + dy)

Example:

dy/dx = (x² + y²) / (2xy)

Solution Method:
Substitution like y = vx or x = vy is commonly used.

Used in:

• Fluid flow

• Symmetric systems
  
  

6. Non-Homogeneous Differential Equation

A Non-Homogeneous Differential Equation has terms of different degrees or an additional function not related to the variable terms.

Example:

 dy/dx + y = sin(x)

• Right-hand side is a function of x only.

• Not all terms are homogeneous.

Used in:

• Forced oscillations

• External influences in systems

• Real-life engineering problems

 

7. Exact Differential Equation

A differential equation of the form:

M(x, y)dx + N(x, y)dy = 0

 is exact if

∂M/∂y = ∂N/∂x

Example:

 (2xy + y²)dx + (x² + 2xy)dy = 0

• ∂M/∂y = 2x + 2y

• ∂N/∂x = 2x + 2y → So it's exact.
  
  

Solution Method:
Find a potential function φ(x, y) such that:

 ∂φ/∂x = M, ∂φ/∂y = N

Used in:

• Thermodynamics

• Potential field problems
  
  

8. Separable Differential Equation

A Separable Differential Equation is one in which the variables can be separated on each side of the equation.

Form:

 dy/dx = g(x) · h(y)

Example:

dy/dx = x²y

Solution Method:
Separate variables:
dy/y = x² dx
Then integrate both sides.

Used in:

• Basic biological models

• Simple dynamic systems

• Chemical reaction rate

 

Summary Table

 

Homogeneous Difference Equation vs. Differential Equation

A homogeneous difference equation is used for discrete variables and recursion.

Example:

aₙ - 3aₙ₋₁ + 2aₙ₋₂ = 0

Though different from a differential equation, it shares similar solving techniques and is widely used in discrete models.

 

Differential Formula: Basic Rules

To solve differential equations, use these common differential formulas:

• d/dx(xⁿ) = nxⁿ⁻¹

• Chain Rule: dy/dx = dy/du × du/dx

• Product Rule: d(uv)/dx = u dv/dx + v du/dx

• Quotient Rule: d(u/v)/dx = (v du/dx - u dv/dx)/v²

These differential formulas are essential tools when solving linear differential equations, partial differential equations, and more.

 

Applications of Differential Equations

The following situations make use of differential equations:

• Physics (electricity, motion, oscillations)
• Biology (disease transmission, population growth)
• Reaction rates in chemistry
• Economics (investment, growth models)
• Engineering (fluid flow, signal processing)

Complex systems with several variables are modelled by partial differential equations.

 

Solved Problems on Differential Equations

Problem 1: Solve dy/dx = 2x

Integrating both sides:
∫dy = ∫2x dx
y = x² + C

 

Problem 2: Solve dy/dx + y = eˣ

This is a linear differential equation.
Integrating factor: eˣ
Multiply: eˣ dy/dx + eˣ y = e²ˣ
LHS becomes: d(yeˣ)/dx = e²ˣ
Integrate: yeˣ = (1/2)e²ˣ + C
Solution: y = (1/2)eˣ + Ce⁻ˣ

 

Problem 3: Solve the partial differential equation ∂u/∂x + ∂u/∂y = 0

General solution: u(x, y) = f(x - y)

 

Problem 4: Solve dy/dx = (x + y)/(x - y)

This is a homogeneous differential equation.
Let y = vx → dy/dx = v + x dv/dx
Substitute and solve using variable separable method.

Conclusion

One essential mathematical tool for understanding systems that change over time is a differential equation. The key to solving theoretical and practical problems is understanding what a differential equation is, how to use differential formulas, and how to differentiate between linear, homogeneous, and partial differential equations.

Differential equations are used to model gravity, population, and financial growth, among other things.

 

Frequently Asked Questions on Differential Equation

1. Which is a differential equation?

Answer: A differential equation is a mathematical equation that relates a function with its derivatives. It shows how a quantity changes over time or space.

Example:
dy/dx + y = 0 is a differential equation because it contains the derivative dy/dx of the function y with respect to x.

 

2. What is ODE and PDE?

Answer:

• ODE (Ordinary Differential Equation):
  A differential equation containing derivatives with respect to only one independent variable.
  Example: dy/dx = x + y

• PDE (Partial Differential Equation):
  A differential equation containing partial derivatives of a function of two or more independent variables.
  Example: ∂²u/∂x² + ∂²u/∂y² = 0

 

3. Is the differential equation in class 11 or 12?

Answer:Differential equations are typically taught in Class 12 as part of the CBSE/ICSE mathematics curriculum in India.
They are covered in the Chapter: Differential Equations under calculus. Basic ideas of derivatives start in Class 11, but solving differential equations is introduced in Class 12.

 

4. What are two types of differential equations?

Answer: The two main types of differential equations are:

• Ordinary Differential Equation (ODE): Involves derivatives with respect to one independent variable only.
  Example: dy/dx + y = 0

• Partial Differential Equation (PDE): Involves partial derivatives with respect to two or more variables.
  Example: ∂u/∂x + ∂u/∂y = 0
  
  

5. What is an example of a differential equation?

Answer:

• One common example of a differential equation is:
  dy/dx = 3x²
• This equation tells us that the rate of change of y with respect to x is equal to 3x². When we solve it, we get the function:
  y = x³ + C, where C is an integration constant.

 

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