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What is the Particular 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?
A particular solution of a first-order linear differential equation is a specific solution that satisfies the equation and also meets certain initial conditions. Unlike the general solution which has an arbitrary constant (like 'C'), the particular solution has a fixed numerical value for this constant. It's like finding a specific route for your auto-rickshaw when you know both the starting point and the destination.
Simple Example
Quick Example
Imagine you are tracking how much mobile data you use. A general solution might tell you your data usage changes by 1GB per day, but it doesn't say how much you started with. If you know you started with 10GB on Monday, then you can find a particular solution for your data usage on any specific day. This specific starting point helps you pinpoint an exact answer.
Worked Example
Step-by-Step
Let's find the particular solution for the differential equation dy/dx + y = 2, given the initial condition y(0) = 1.
1. This is a first-order linear differential equation of the form dy/dx + P(x)y = Q(x), where P(x) = 1 and Q(x) = 2.
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2. Find the integrating factor (IF). IF = e^(integral of P(x) dx) = e^(integral of 1 dx) = e^x.
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3. Multiply the entire differential equation by the integrating factor: e^x (dy/dx + y) = 2e^x. This simplifies to d/dx (y * e^x) = 2e^x.
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4. Integrate both sides with respect to x: integral of d/dx (y * e^x) dx = integral of 2e^x dx.
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5. This gives y * e^x = 2e^x + C, where C is the constant of integration. This is the general solution.
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6. Now, use the initial condition y(0) = 1. Substitute x = 0 and y = 1 into the general solution: 1 * e^0 = 2e^0 + C.
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7. Since e^0 = 1, we get 1 * 1 = 2 * 1 + C, which means 1 = 2 + C. Solving for C, we get C = 1 - 2 = -1.
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8. Substitute C = -1 back into the general solution: y * e^x = 2e^x - 1. Therefore, the particular solution is y = (2e^x - 1) / e^x or y = 2 - e^(-x).
Why It Matters
Particular solutions are vital in fields like AI/ML and Physics. Engineers use them to predict the exact path of a rocket (Space Technology) or model how a specific medicine spreads in the body (Medicine). Understanding these helps you design better technology and make accurate predictions, opening doors to careers in research and development.
Common Mistakes
MISTAKE: Forgetting to find the constant C using the initial condition. | CORRECTION: Always use the given initial condition (e.g., y(0)=1) to calculate the specific value of C after finding the general solution.
MISTAKE: Incorrectly calculating the integrating factor. | CORRECTION: Double-check your integration of P(x) and the exponentiation for e^(integral P(x) dx). A small error here will make the whole solution wrong.
MISTAKE: Mixing up the general solution with the particular solution. | CORRECTION: Remember, the general solution has 'C' in it, while the particular solution has a specific number instead of 'C' because you've used the initial condition.
Practice Questions
Try It Yourself
QUESTION: Find the particular solution for dy/dx + 2y = 4, given y(0) = 3. | ANSWER: y = 2 + e^(-2x)
QUESTION: Find the particular solution for dy/dx + (1/x)y = x, given y(1) = 0. | ANSWER: y = (x^2)/3 - 1/(3x)
QUESTION: A population of bacteria grows according to the differential equation dP/dt = 0.5P, where P is the population at time t. If the initial population P(0) = 100, find the particular solution for P(t). | ANSWER: P(t) = 100e^(0.5t)
MCQ
Quick Quiz
What is the main difference between a general solution and a particular solution of a differential equation?
A particular solution is always simpler to calculate.
A general solution includes an arbitrary constant 'C', while a particular solution has a specific numerical value for 'C'.
A general solution only applies to first-order equations.
A particular solution has no variables, only numbers.
The Correct Answer Is:
B
The general solution includes an arbitrary constant 'C' because there are infinitely many solutions. A particular solution uses an initial condition to find a specific value for 'C', thus giving a unique solution.
Real World Connection
In the Real World
Imagine you are building a new electric vehicle (EV) in India. To model how its battery drains over time, engineers use differential equations. A general solution might tell you the pattern of discharge. But to know exactly how long a specific EV battery, starting with a full charge, will last before needing a recharge, you need a particular solution. This helps companies like Tata Motors or Mahindra design efficient EVs.
Key Vocabulary
Key Terms
DIFFERENTIAL EQUATION: An equation involving derivatives of an unknown function.| FIRST ORDER: An equation where the highest derivative is the first derivative.| LINEAR: An equation where the dependent variable and its derivatives appear only to the first power.| INITIAL CONDITION: A specific value of the function at a given point, used to find the constant C.| INTEGRATING FACTOR: A special function used to solve first-order linear differential equations.
What's Next
What to Learn Next
Great job understanding particular solutions! Next, you should explore 'Higher Order Linear Differential Equations.' This will build on what you've learned by introducing equations with second or even higher derivatives, opening up even more complex and interesting real-world problems.
