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What is the General Solution of First Order Linear Differential Equations?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The general solution of a first-order linear differential equation is a formula that gives ALL possible solutions to that equation. It includes an arbitrary constant, 'C', because there are infinitely many specific solutions that fit the pattern.
Simple Example
Quick Example
Imagine you know how fast a delivery scooter's speed changes (its acceleration) over time. A first-order linear differential equation helps describe this. The general solution would be a formula that tells you the scooter's speed at any given time, but it would have a 'starting speed' that you can change. This 'starting speed' is like the constant 'C'.
Worked Example
Step-by-Step
Let's find the general solution for the equation: dy/dx + 2y = 4.
Step 1: Identify P(x) and Q(x). Here, P(x) = 2 and Q(x) = 4.
---Step 2: Calculate the Integrating Factor (IF) using the formula e^(integral P(x) dx). So, IF = e^(integral 2 dx) = e^(2x).
---Step 3: Multiply the entire differential equation by the Integrating Factor. This gives: e^(2x) * (dy/dx + 2y) = 4 * e^(2x).
---Step 4: The left side becomes d/dx (y * IF). So, d/dx (y * e^(2x)) = 4 * e^(2x).
---Step 5: Integrate both sides with respect to x. integral [d/dx (y * e^(2x))] dx = integral [4 * e^(2x)] dx.
---Step 6: This simplifies to y * e^(2x) = 4 * (e^(2x) / 2) + C.
---Step 7: Simplify further: y * e^(2x) = 2 * e^(2x) + C.
---Step 8: Isolate y to get the general solution: y = (2 * e^(2x) + C) / e^(2x) which simplifies to y = 2 + C * e^(-2x).
Answer: The general solution is y = 2 + C * e^(-2x).
Why It Matters
Understanding general solutions helps engineers design electric vehicles (EVs) by predicting battery discharge, or doctors model how medicine spreads in the body. If you want to work in AI/ML, FinTech, or even space technology, this concept is a building block for solving real-world problems and creating new technologies.
Common Mistakes
MISTAKE: Forgetting to add the constant 'C' after integration. | CORRECTION: Always remember to add '+ C' when you perform indefinite integration, as it represents the family of all possible solutions.
MISTAKE: Incorrectly calculating the Integrating Factor (IF), especially with signs or complex P(x). | CORRECTION: Double-check the integral of P(x) in the exponent of 'e'. Pay close attention to negative signs and variable functions.
MISTAKE: Not correctly isolating 'y' at the end or making algebraic errors while doing so. | CORRECTION: After integrating, make sure to divide the entire right-hand side by the Integrating Factor to express 'y' explicitly.
Practice Questions
Try It Yourself
QUESTION: Find the general solution of dy/dx + y = 1. | ANSWER: y = 1 + C * e^(-x)
QUESTION: Find the general solution of dy/dx + (y/x) = x^2. | ANSWER: y = (x^3 / 4) + (C / x)
QUESTION: A population 'P' grows at a rate such that dP/dt + 2P = 10. Find the general solution for P(t). | ANSWER: P(t) = 5 + C * e^(-2t)
MCQ
Quick Quiz
What is the integrating factor for the differential equation dy/dx + (3/x)y = sin(x)?
e^(3x)
x^3
3x
e^(3/x)
The Correct Answer Is:
B
The integrating factor is e^(integral P(x) dx). Here P(x) = 3/x, so integral (3/x) dx = 3 ln|x| = ln(x^3). Therefore, the integrating factor is e^(ln(x^3)) which simplifies to x^3.
Real World Connection
In the Real World
Imagine a water tank in an Indian village that is constantly being filled by a pump and also has water being used for daily needs. A first-order linear differential equation can model how the water level changes over time. The general solution would tell us all possible water levels, depending on the initial amount of water in the tank. This helps engineers plan water supply systems or predict flood situations.
Key Vocabulary
Key Terms
DIFFERENTIAL EQUATION: An equation involving derivatives of a function | FIRST-ORDER: Involves only the first derivative (dy/dx) | LINEAR: The dependent variable 'y' and its derivatives appear only with power 1 and are not multiplied together | INTEGRATING FACTOR: A special function used to solve linear differential equations | GENERAL SOLUTION: A solution that contains an arbitrary constant 'C' and represents all possible solutions
What's Next
What to Learn Next
Great job understanding general solutions! Next, you should learn about 'Particular Solutions of First Order Linear Differential Equations'. This will show you how to use extra information, like a starting value, to find a single, unique solution from the general solution you just learned.
