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What is the Particular Solution of First Order 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 particular solution of a first-order differential equation is a specific solution obtained by using given initial conditions. Unlike a general solution that has an arbitrary constant 'C', a particular solution has a fixed numerical value for 'C', making it unique.
Simple Example
Quick Example
Imagine you know the general formula for how your mobile data usage changes each hour. If you also know you started with 10 GB of data at 9 AM, you can find a particular formula that describes exactly how much data you have at any specific time, rather than a general one that works for anyone. This 'starting condition' helps find your unique data usage pattern.
Worked Example
Step-by-Step
Let's find the particular solution for the differential equation dy/dx = 2x, given that y(1) = 5.
1. First, find the general solution by integrating the differential equation:
dy = 2x dx
∫dy = ∫2x dx
y = x^2 + C
---2. Now, use the initial condition y(1) = 5. This means when x = 1, y = 5. Substitute these values into the general solution:
5 = (1)^2 + C
---3. Solve for C:
5 = 1 + C
C = 5 - 1
C = 4
---4. Substitute the value of C back into the general solution to get the particular solution:
y = x^2 + 4
Answer: The particular solution is y = x^2 + 4.
Why It Matters
Understanding particular solutions helps engineers design electric vehicle batteries to last a specific time, or physicists predict the exact path of a rocket launched by ISRO. Financial analysts also use it to model specific stock price movements based on initial market conditions, leading to careers in FinTech and Space Technology.
Common Mistakes
MISTAKE: Forgetting to integrate the differential equation first and directly substituting initial conditions. | CORRECTION: Always find the general solution (which includes the constant 'C') by integrating the differential equation before using the initial conditions.
MISTAKE: Substituting the initial conditions into the original differential equation instead of the general solution. | CORRECTION: The initial conditions (like y(x0) = y0) are used to find the value of 'C' in the integrated general solution, not the original equation.
MISTAKE: Making calculation errors when solving for 'C'. | CORRECTION: Double-check your arithmetic when substituting the values of x and y from the initial condition and solving the resulting equation for 'C'.
Practice Questions
Try It Yourself
QUESTION: Find the particular solution of dy/dx = 3x^2, given y(0) = 1. | ANSWER: y = x^3 + 1
QUESTION: Find the particular solution of dy/dx = 2e^(2x), given y(0) = 3. | ANSWER: y = e^(2x) + 2
QUESTION: The rate of change of temperature T (in degrees Celsius) of a cooling cup of chai with respect to time t (in minutes) is given by dT/dt = -0.5. If the initial temperature of the chai is 80 degrees Celsius at t=0, find the particular solution for T(t). | ANSWER: T(t) = -0.5t + 80
MCQ
Quick Quiz
What is needed to find a particular solution from a general solution of a first-order differential equation?
Another differential equation
An arbitrary constant 'C'
Initial conditions or boundary conditions
The derivative of the general solution
The Correct Answer Is:
C
Initial conditions provide specific values for x and y that allow us to find a unique numerical value for the arbitrary constant 'C' in the general solution, thus leading to a particular solution. Options A, B, and D are incorrect as they don't serve this purpose.
Real World Connection
In the Real World
In India, meteorologists use particular solutions to predict specific weather patterns. For example, if they have a general model for how temperature changes, and they know today's temperature and humidity (initial conditions), they can find a particular solution to forecast tomorrow's temperature for your city, helping farmers plan their crops or people plan their day.
Key Vocabulary
Key Terms
DIFFERENTIAL EQUATION: An equation involving derivatives of a function | GENERAL SOLUTION: A solution to a differential equation that includes an arbitrary constant 'C' | INITIAL CONDITIONS: Specific values of the variables at a given point, used to find 'C' | ARBITRARY CONSTANT: A constant 'C' that can take any value in a general solution
What's Next
What to Learn Next
Great job understanding particular solutions! Next, you can explore 'Higher Order Differential Equations'. This will build on your current knowledge by showing how similar principles apply when equations involve second or third derivatives, which are crucial for more complex real-world models.
