Case Study Differential Equations Class 12

Case Study Differential Equations Class 12

Case Study Differential Equations Class 12 — Free Online Test

Case Study Differential Equations Class 12 — Free Online Test

The Case Study Differential Equations Class 12 free online test helps students master core concepts through application-based problems. It includes math case study questions designed to improve analytical and problem-solving skills. Because it follows the CBSE pattern, the content is perfect for class 12 math case study questions.

Structured Practice for Class 12 Students

Focusing on math case study questions for class 12, the test covers linear and nonlinear differential equations. Moreover, each problem includes a step-by-step solution. This approach helps students identify mistakes and learn correct methods quickly.

Exam-Oriented Learning

The Case Study Differential Equations Class 12 test builds accuracy and speed. Therefore, students gain confidence before exams. It also improves the ability to solve class 12 math case study questions under time constraints.

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 \( y = vx \Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx} \).

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