Problem 11 - A Complete Resource Book in Mathematics for JEE Main

Question

If the function \(y(x)\) represented by \(x=\sin t\) \(y=a e^{l \sqrt{2}}+b e^{l \sqrt{2}}, t \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) satisfies the equation \(\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}=k y\), then \(k\) is equal to (A) 1 (B) \(-2\) (C) 2 (D) None of these

Step-by-Step Solution

Answer

k is 0 for the equation to hold with the given conditions.

Step 1: Understand the Components

The given function representation is in parameterized form with \(x = \sin t\). The function \(y\) is given by \(y = ae^{l\sqrt{2}} + be^{l\sqrt{2}}\). The goal is to find the value of \(k\) in the differential equation \((1-x^2)y'' - xy' = ky\).

Step 2: Analyze the Form of y(x)

The function \(y\) seems to be constant because it's not presented as a function of \(t\), but as a sum of constants multiplied by the same factor. This indicates \(y' = 0\) and \(y'' = 0\) since \(a\), \(b\), and \(l\) are constants.

Step 3: Substitute into Differential Equation

When \(y' = 0\) and \(y'' = 0\), substitute into the differential equation \((1-x^2)y'' - xy' = ky\), which simplifies to \(0 - 0 = ky\). This leads us to \(0 = ky\).

Step 4: Solve for k

Since \(y eq 0\) under normal conditions, for the equation \(0 = ky\) to hold, it must be true that \(k = 0\). This is required for the left-hand side to remain zero.

Key Concepts

Parameterized Function

In mathematics, a parameterized function allows us to describe a curve or a surface using an independent variable known as a parameter. This way of representation is particularly useful in situations where direct expression in terms of one variable is complex or impossible.

For example, the given exercise involves a parameterized function where the curve is described using the parameter \(t\). Here, \(x = \sin t\), which parameterizes \(x\) as a function of \(t\). This representation allows us to explore different values of \(x\) by changing \(t\) within the specified range \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Parameterized functions are versatile as they offer:

• A way to simplify calculations and expressions for curves and surfaces.
• The ability to work with implicit relationships between variables.
• Aids in finding the derivative of a function with respect to the parameter, thus facilitating the study of the behavior of a function graphically.

Understanding parameterization is key when solving complex differential equations involving such functions.

Solution of Differential Equation

A differential equation is an equation involving derivatives of a function. Finding its solution means finding a function that satisfies the equation. In our example, the function must satisfy the equation \((1-x^{2})y'' - x y' = k y\).

Initially, it might seem that solving such equations requires intensive calculations. However, the step-by-step process illustrates a simpler way when specific conditions, such as constants in the given function, apply:

• Recognizing that in some cases, the function \(y(x)\) can be considered a constant with respect to the parameter simplifies the equation.
• Substituting the derivatives into the equation then results in straightforward expressions to solve for the unknown parameters, like \(k\).

Solving differential equations is vital in understanding various phenomena, since these equations appear in many areas such as physics, engineering, and biology. Successfully solving a differential equation involves discovering patterns or constants that keep the system balanced under given conditions.

Mathematical Problem-Solving Steps

Problem-solving in mathematics often requires a structured approach to understand complex concepts and derive solutions. The step-by-step method utilized in the exercise emphasizes the importance of breaking down problems methodically.

Here is how you can apply these steps effectively:

• **Understand the Components:** Analyze and comprehend the various elements of the given equation or problem statement. Identify what each symbol and expression represents.
• **Analyze Simplifications:** Simplify your working by identifying constants and recognizing terms that reduce complexity substantially.
• **Substitute Methodically:** Carefully substitute simplified terms back into the equation. This helps in verifying if further simplification or cancellation can occur.
• **Solve Strategically:** Look for insights such as constants or relationships among variables that balance the equation. Use this to arrive at the solution comprehensively.

By applying these structured problem-solving steps, students can gain a deeper understanding and develop a stronger capacity for tackling challenging mathematical problems.