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What is the First Order Linear Differential Equation?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
A First Order Linear Differential Equation is a special type of equation that involves a function and its first derivative. 'First order' means it only has the first derivative (like dy/dx), and 'linear' means the function and its derivative are not multiplied together or raised to powers. It helps us understand how things change over time or space in a straightforward way.
Simple Example
Quick Example
Imagine the speed at which your mobile phone battery drains. If the rate of draining (dy/dt) depends directly on how much charge is left (y) and also on some constant factor (like background apps), that relationship can often be described by a first order linear differential equation. It helps predict how long your battery will last!
Worked Example
Step-by-Step
Let's solve a simple first order linear differential equation: dy/dx + y = 2
Step 1: Identify P(x) and Q(x). Here, P(x) = 1 (coefficient of y) and Q(x) = 2.
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Step 2: Calculate the Integrating Factor (IF). IF = e^(integral P(x) dx). So, IF = e^(integral 1 dx) = e^x.
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Step 3: Multiply the entire equation by the Integrating Factor. e^x (dy/dx) + e^x (y) = 2e^x.
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Step 4: The left side is now the derivative of (y * IF). So, d/dx (y * e^x) = 2e^x.
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Step 5: Integrate both sides with respect to x. integral [d/dx (y * e^x)] dx = integral [2e^x] dx.
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Step 6: This gives y * e^x = 2e^x + C, where C is the constant of integration.
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Step 7: Solve for y. Divide by e^x. y = (2e^x + C) / e^x.
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Step 8: Simplify. y = 2 + C * e^(-x).
Answer: The solution is y = 2 + C * e^(-x).
Why It Matters
These equations are like secret codes for predicting how things change! Engineers use them to design electric vehicles (EVs) and understand how batteries charge. Doctors use them to model how medicines spread in the body. Even AI/ML algorithms use similar math to learn patterns from data.
Common Mistakes
MISTAKE: Forgetting the constant of integration 'C' when integrating. | CORRECTION: Always add '+ C' after performing indefinite integration, as it represents a family of solutions.
MISTAKE: Incorrectly calculating the Integrating Factor, especially when P(x) is complex. | CORRECTION: Double-check the integration of P(x) and ensure the exponential function is applied correctly.
MISTAKE: Not separating variables properly or mixing up which term is P(x) and which is Q(x). | CORRECTION: Always ensure the equation is in the standard form dy/dx + P(x)y = Q(x) before identifying P(x) and Q(x).
Practice Questions
Try It Yourself
QUESTION: Find the integrating factor for the differential equation dy/dx + (2/x)y = x^2. | ANSWER: e^(integral (2/x) dx) = e^(2 ln|x|) = e^(ln|x|^2) = x^2
QUESTION: Solve the differential equation dy/dx + y = 1, given y(0) = 3. | ANSWER: y = 1 + 2e^(-x)
QUESTION: The rate at which a cup of chai cools is proportional to the difference between its temperature and the room temperature. If the room temperature is 20 degrees Celsius and the initial temperature of chai is 80 degrees Celsius, set up a first order linear differential equation to model its cooling. (Do not solve). | ANSWER: Let T be the temperature of chai and t be time. Then dT/dt = k(T - 20), where k is a constant. Rearranging: dT/dt - kT = -20k.
MCQ
Quick Quiz
Which of the following is a First Order Linear Differential Equation?
dy/dx + y^2 = x
d^2y/dx^2 + y = 0
dy/dx + x*y = sin(x)
(dy/dx)^2 + y = x
The Correct Answer Is:
C
Option C is in the standard form dy/dx + P(x)y = Q(x), where P(x) = x and Q(x) = sin(x). Options A and D are not linear due to y^2 and (dy/dx)^2 respectively. Option B is second order due to d^2y/dx^2.
Real World Connection
In the Real World
Imagine you are an engineer at ISRO designing a satellite. You need to predict how quickly its internal temperature changes when it enters sunlight or shadow. First order linear differential equations help model this heat transfer, ensuring the sensitive equipment stays safe. Similarly, FinTech experts use them to model how investment values change over time.
Key Vocabulary
Key Terms
DIFFERENTIAL EQUATION: An equation involving derivatives of an unknown function | FIRST ORDER: Involves only the first derivative (like dy/dx) | LINEAR: The function and its derivatives appear only to the power of one and are not multiplied together | INTEGRATING FACTOR: A special function used to solve first order linear differential equations | DERIVATIVE: The rate at which one quantity changes with respect to another
What's Next
What to Learn Next
Great job understanding first order linear differential equations! Next, you can explore 'Applications of Differential Equations' to see how these equations are used in real-world problems like population growth, radioactive decay, and circuit analysis. This will show you the power of what you just learned!
