Problem 74 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(y=c_{1} e^{2 x}+c_{2} e^{x}+c_{3} e^{-x}\) satisfies the differential equation \(\frac{d^{3} y}{d x^{3}}+a \frac{d^{2} y}{d x^{2}}+b \frac{d y}{d x}+c y=0\), then \(\frac{a^{3}+b^{3}+c^{3}}{a b c}\) is equal to (A) \(\frac{1}{4}\) (B) \(-\frac{1}{4}\) (C) \(\frac{1}{2}\) (D) \(-\frac{1}{2}\)

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{1}{2}\), choice (D).

Step 1: Understand Differential Equation

We are given that the function \( y = c_1 e^{2x} + c_2 e^{x} + c_3 e^{-x} \) satisfies the differential equation \( \frac{d^3 y}{dx^3} + a \frac{d^2 y}{dx^2} + b \frac{d y}{dx} + c y = 0 \). Here, \( a, b, \) and \( c \) are constants.

Step 2: Compute Derivatives of y

Calculate the first, second, and third derivatives of \( y \): \( \frac{dy}{dx} = 2c_1 e^{2x} + c_2 e^{x} - c_3 e^{-x} \) \( \frac{d^2y}{dx^2} = 4c_1 e^{2x} + c_2 e^{x} + c_3 e^{-x} \) \( \frac{d^3y}{dx^3} = 8c_1 e^{2x} + c_2 e^{x} - c_3 e^{-x} \).

Step 3: Substitute Derivatives into Differential Equation

Substitute the computed derivatives into the differential equation: \( 8c_1e^{2x} + c_2e^{x} - c_3e^{-x} + a(4c_1e^{2x} + c_2e^{x} + c_3e^{-x}) + b(2c_1e^{2x} + c_2e^{x} - c_3e^{-x}) + c(c_1e^{2x} + c_2e^{x} + c_3e^{-x}) = 0 \).

Step 4: Combine Like Terms by Coefficients

Combine the exponential terms based on exponents from the substituted equation: For \( e^{2x} \): \( (8 + 4a + 2b + c)c_1 \) For \( e^{x} \): \( (1 + a + b + c)c_2 \) For \( e^{-x} \): \( (-1 + a - b + c)c_3 \). All coefficients must equal zero.

Step 5: Solve the System of Equations

Set each coefficient equal to zero: 1. \( 8 + 4a + 2b + c = 0 \) 2. \( 1 + a + b + c = 0 \) 3. \( -1 + a - b + c = 0 \). Solve for \( a, b, \) and \( c \).

Step 6: Solve for a, b, c

From these equations, find consistent values of \( a, b, \) and \( c \): Solve (1) for c: \( c = -8 - 4a - 2b \) Substitute in (2) and (3), solve the system of equations to get \( a = 1, b = -2, c = -3 \).

Step 7: Calculate the Expression \( \frac{a^3 + b^3 + c^3}{abc} \)

Substitute the values \( a = 1, b = -2, c = -3 \) into the expression \( \frac{a^3 + b^3 + c^3}{abc} \): \( a^3 = 1, b^3 = -8, c^3 = -27 \) \( abc = (1)(-2)(-3) = 6 \) Plug values into the expression: \[ \frac{1 - 8 - 27}{6} = \frac{-34}{6} = -\frac{17}{3} \].

Key Concepts

Exponential Functions

Exponential functions are an integral part of mathematics, characterized by a constant base raised to a variable exponent. These functions can be represented generally as \( y = a \cdot e^{kx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. In the given function \( y = c_1 e^{2x} + c_2 e^{x} + c_3 e^{-x} \), each term is an exponential function.

• \( e^{2x} \) indicates exponential growth, as the exponent \( 2x \) increases with \( x \).
• \( e^{x} \) also shows growth but at a slower rate compared to \( e^{2x} \).
• \( e^{-x} \) represents exponential decay, where the function value decreases as \( x \) increases.

Exponential functions are incredibly useful in modeling real-world phenomena, such as population growth and radioactive decay. Understanding these properties helps us analyze and solve complex differential equations like the one given in the exercise.

Higher Order Derivatives

Higher order derivatives involve taking the derivative of a function multiple times. In this context, the third derivative \( \frac{d^3 y}{dx^3} \) is of interest. Computing higher order derivatives allows us to examine the behavior and changes in the slope of the original function.

Here, the process of obtaining derivatives from \( y = c_1 e^{2x} + c_2 e^{x} + c_3 e^{-x} \) is executed sequentially:

• First derivative: \( \frac{dy}{dx} = 2c_1 e^{2x} + c_2 e^{x} - c_3 e^{-x} \)
• Second derivative: \( \frac{d^2y}{dx^2} = 4c_1 e^{2x} + c_2 e^{x} + c_3 e^{-x} \)
• Third derivative: \( \frac{d^3y}{dx^3} = 8c_1 e^{2x} + c_2 e^{x} - c_3 e^{-x} \)

Each derivative involves applying the rules of differentiation to exponential functions, adjusting their coefficients and exponents accordingly. This skill is essential when working with differential equations to ensure a correct and accurate solution.

System of Equations

Solving a system of equations is a crucial step when multiple equations need to hold true simultaneously. In the given problem, after substituting the derivatives into the differential equation, we combine the terms by their exponential coefficients. The condition for the original differential equation to hold is that coefficients of like terms equal zero, leading to a system of linear equations:

• \( 8 + 4a + 2b + c = 0 \)
• \( 1 + a + b + c = 0 \)
• \( -1 + a - b + c = 0 \)

By solving this system of equations, we aim to find values of \( a, b, \) and \( c \) that satisfy all the equations. It involves substitution or elimination methods to systematically reduce the equations and extract valid solutions for the constants.

Understanding this procedure is essential for deriving steady-state solutions or ensuring continuity and consistency across different mathematical models, a common requirement in engineering and physical sciences.