Problem 94 - A Complete Resource Book in Mathematics for JEE Main

Question

In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is acorrect explanation forAssertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: The order of the differential equation, of which \(x y=c e^{x}+b e^{-x}+x^{2}\) is a solution, is 2 . Reason: The differential equation is \(x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}-x y+x^{2}-2=0\)

Step-by-Step Solution

Answer

Option A: Both Assertion and Reason are true, and the reason is the correct explanation for the assertion.

Step 1: Understanding the assertion

The assertion states that the order of the differential equation, which has the solution \(xy = ce^{x} + be^{-x} + x^{2}\), is 2.

Step 2: Analyze the Reason

The reason provides a specific differential equation: \(x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0\). We need to determine the order of this given differential equation.

Step 3: Determine the Order of the Differential Equation

The order of a differential equation is defined as the highest derivative present in the equation. In the equation \(x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0\), the highest derivative is \(\frac{d^{2} y}{d x^{2}}\), which is a second derivative.

Step 4: Verify if the Reason explains the Assertion

Since the differential equation provided in the reason has a second derivative as its highest derivative, its order is 2. This matches the assertion, which claims the order of the differential equation is 2. Therefore, the reason correctly explains why the assertion is true.

Key Concepts

Order of Differential Equations

The order of a differential equation refers to the highest derivative present in the equation. This is an important concept because it helps determine the complexity and nature of the solutions that the equation might have. For example, if a differential equation has a second derivative like \(\frac{d^2 y}{dx^2}\), then its order is 2. Such an equation typically models systems with acceleration, such as those found in physics related to motion, because acceleration is a second derivative of position with respect to time.

To identify the order:

• Look for the term with the derivative having the highest order.
• The coefficient of this derivative term doesn't affect the order.
• The order is purely dependent on the highest derivative itself.

Understanding the order helps in exploring the potential solutions and the techniques required to solve these differential equations effectively.

Assertion and Reason in Mathematics

Assertions and reasons form a key part of logical reasoning in mathematics. An assertion is a statement that claims a specific fact is true. It can appear as a standalone statement or be paired with a reason, which provides a rationale supporting why the assertion is believed to be true. In questions like the one given, evaluating both the assertion and the reason together is crucial.

Here's how they work:

• An assertion might not always be directly linked to the reason.
• The reason is considered sufficient if it fully and correctly explains why the assertion is true.
• Sometimes, both the assertion and the reason are true independently, but they don't necessarily explain each other.

When you encounter these in exercises, check if the explanation (reason) truly elucidates why the assertion is correct. This enhances critical thinking and comprehension of mathematical concepts.

Solution of Differential Equations

Solving differential equations involves finding a function or functions that satisfy the conditions of the equation. This is central to a variety of scientific and engineering disciplines as it allows us to understand and predict behavior within dynamical systems. Solutions can be general or particular, often involving constants determined by initial conditions.

Here are steps to solve them:

• Identify the order and type of differential equation.
• Choose an appropriate method such as separation of variables, integration, or using specific formulas.
• For higher-order equations, calculate derivatives or employ characteristic equations.
• Accurately apply boundary or initial conditions to find particular solutions.

Understanding the process of solving differential equations enables you to tackle real-world problems such as predicting population growth, heat transfer, or financial modeling effectively.