Mathematics Exam > Mathematics Videos > Calculus > First Order Linear Differential Equations
First Order Linear Differential Equations Video Lecture | Calculus - Mathematics
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FAQs on First Order Linear Differential Equations Video Lecture - Calculus - Mathematics
| 1. What is a first-order linear differential equation? |  |
Ans. A first-order linear differential equation is a differential equation in which the highest power of the derivative is 1, and the equation is linear in terms of the dependent variable and its derivative. It can be written in the form: dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x.
| 2. How do you solve a first-order linear differential equation? | |
Ans. To solve a first-order linear differential equation, you can use the method of integrating factors. First, rewrite the equation in the standard form: dy/dx + P(x)y = Q(x). Then, find the integrating factor, which is the exponential of the integral of P(x) with respect to x. Multiply both sides of the equation by the integrating factor and integrate both sides to obtain the solution.
| 3. What is the role of integrating factors in solving first-order linear differential equations? | |
Ans. Integrating factors play a crucial role in solving first-order linear differential equations. They help transform the equation into an exact differential equation, making it easier to integrate. The integrating factor is used to multiply both sides of the equation, which allows it to be rewritten in a form that can be integrated directly.
| 4. Can you provide an example of solving a first-order linear differential equation? | |
Ans. Sure! Let's consider the equation dy/dx + 2x^2y = x^2. First, we rewrite it in the standard form: dy/dx + 2x^2y = x^2. Then, we find the integrating factor, which is e^(∫2x^2dx) = e^(2/3x^3). Multiply both sides of the equation by the integrating factor to get e^(2/3x^3)dy/dx + 2x^2e^(2/3x^3)y = x^2e^(2/3x^3). Integrating both sides, we have ∫e^(2/3x^3)dy/dx dx + ∫2x^2e^(2/3x^3)y dx = ∫x^2e^(2/3x^3)dx. After integrating and simplifying, we get y = C*e^(-2/3x^3) + (1/3)e^(-2/3x^3)x^3 + (1/9)e^(-2/3x^3)x^6, where C is the constant of integration.
| 5. What are some real-life applications of first-order linear differential equations? | |
Ans. First-order linear differential equations have various applications in real life. They are used to model population growth, radioactive decay, cooling of objects, drug concentration in a patient's body, and many other physical and biological phenomena. By solving these differential equations, we can understand and predict how these processes change over time.
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Nov 22, 2024 Last updated |
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