Case Study Differential Equations Class 12

Case Study Differential Equations Class 12

Chapter: Case Study Differential Equations Class 12

The chapter on Case Study Differential Equations Class 12 introduces students to real-life applications of differential equations. Moreover, it helps them understand how mathematical models are formed using order, degree, and basic solution techniques. Therefore, learners can apply formulas confidently while solving application-based questions.

Understanding Core Concepts

This section explains the formation of differential equations and their practical uses. Additionally, students explore multiple solution methods that simplify analysis. These concepts are essential for handling case study questions effectively.

Benefits of Practicing Case Studies

Practicing case studies builds strong analytical skills. Consequently, students understand how differential equations relate to real situations. This approach not only improves logical reasoning but also prepares learners well for Class 12 board examinations.

Case Study 3

A company is studying the temperature variation in a metal rod heated at one end and insulated at the other. The temperature \( y \) at a point \( x \) along the rod is governed by the differential equation:

\[ \frac{dy}{dx} + Py = Q \]

This is a linear differential equation. Solving such equations involves finding the integrating factor (IF), which is given by:

\[ \text{IF} = e^{\int P dx} \]

The general solution is:

\[ y \cdot \text{IF} = \int Q \cdot \text{IF} \, dx + C \]

Other types of solvable first-order differential equations include:

  • Variables separable: \( \frac{dy}{dx} = f(x)g(y) \)
  • Homogeneous: \( \frac{dy}{dx} = F\left( \frac{y}{x} \right) \), solved using substitution \( y = vx \)

Engineers often model heat flow, current in circuits, or population changes using these techniques.

MCQ Questions

1. The standard form of a linear differential equation is:
Solution: A first-order linear differential equation has the form \( \frac{dy}{dx} + Py = Q \), where \( P \) and \( Q \) are functions of \( x \).
2. The integrating factor for \( \frac{dy}{dx} + 3y = 6 \) is:
Solution: Here, \( P = 3 \). So the integrating factor is: \[ IF = e^{\int 3 dx} = e^{3x} \]
3. Solve the differential equation: \[ \frac{dy}{dx} + y \tan x = \sin x \] Which one of the following is the general solution?
Solution:

This is a linear differential equation of the form \( \frac{dy}{dx} + P(x) y = Q(x) \) where \( P(x) = \tan x \), and \( Q(x) = \sin x \).

Step 1: Find the Integrating Factor (IF) \[ \text{IF} = e^{\int P(x)\, dx} = e^{\int \tan x\, dx} = e^{-\ln|\cos x|} = \frac{1}{\cos x} \]

Step 2-4: The general solution is: \[ y = \cos x \left( -\ln|\cos x| + C \right) \]

4. The equation \( \frac{dy}{dx} = \frac{x+y}{x} \) is:
Solution: Rewrite as: \[ \frac{dy}{dx} = 1 + \frac{y}{x} = F\left( \frac{y}{x} \right) \] Substituting \( y = vx \) reduces it to a separable form. Hence it is homogeneous.
5. Which substitution is used to solve a homogeneous differential equation?
Solution: For a homogeneous equation \( \frac{dy}{dx} = F\left( \frac{y}{x} \right) \), use \( y = vx \Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx} \).

Explore More Math Case Studies On Differential Equations class 12

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