Problem 58 - A Textbook of Differential Calculus

Question

Let \(f_{1}(x)\) and \(f_{2}(x)\) be twice differentiable functions where \(F(x)=f_{1}(x)+f_{2}(x)\) and \(G(x)=f_{1}(x)-f_{2}(x), \forall x \in R, f_{1}(0)=2\) and \(f_{2}(0)=1 .\) If \(f_{1}^{\prime}(x)=f_{2}(x)\) and \(f_{2}^{\prime}(x)=f_{1}(x), \forall x \in R\), then the number of solutions of the equation \((F(x))^{2}=\frac{9 x^{4}}{G(x)}\) is .........

Step-by-Step Solution

Answer

The number of solutions is typically finite and primarily concentrated based on specific interval solutions along periodic nature.

Step 1: Write down given relationships

We have:\( F(x) = f_1(x) + f_2(x) \) and \( G(x) = f_1(x) - f_2(x) \). Also, \( f_1'(x) = f_2(x) \) and \( f_2'(x) = f_1(x) \). These differential equations resemble the derivatives of sine and cosine functions.

Step 2: Solve differential equations

The system of equations: \( f_1'(x) = f_2(x) \) and \( f_2'(x) = f_1(x) \) leads us to consider functions of the form \( f_1(x) = a\cos(x) + b\sin(x) \) and \( f_2(x) = a\sin(x) - b\cos(x) \).

Step 3: Apply initial conditions

Using initial conditions \( f_1(0) = 2 \) and \( f_2(0) = 1 \), we solve to find \( a \) and \( b \). By substituting \( x = 0 \), we get \( f_1(0) = a = 2 \) and \( f_2(0) = b = 1 \).

Step 4: Express functions explicitly

With \( a = 2 \) and \( b = 1 \), we express the functions:\( f_1(x) = 2\cos(x) + \sin(x) \) and \( f_2(x) = 2\sin(x) - \cos(x) \).

Step 5: Find expressions for F(x) and G(x)

Calculate \( F(x) = f_1(x) + f_2(x) = 3\sin(x) + \cos(x) \) and \( G(x) = f_1(x) - f_2(x) = 3\cos(x) + \sin(x) \).

Step 6: Simplify the equation (F(x))^2 = 9x^4/G(x)

Substitute \( F(x) \) and \( G(x) \) into the equation:\( (3\sin(x) + \cos(x))^2 = \frac{9x^4}{3\cos(x) + \sin(x)} \).

Step 7: Solve the trigonometric equation approximately

Look for intersections for trigonometric functions:Simplifying both sides of the equation to a common trigonometric form, the main solutions in \[0, \pi\] can be approximated since they correspond to multiples of \(2\pi\) due to periodicity. Analyze such windows for solutions.

Step 8: Conclude with the general solution conditions

Analyze periodic solutions within a period, repeat due to periodicity of sine and cosine functions. For the equation's nature and the complexity of intersections, check specific solutions:

Step 9: Determine number of solutions

The system, when checked, typically yields limited distinct solutions over a principal interval like \([0, 2\pi)\), repeated over periodic intervals. Ultimately, these involve checking real solutions given form, e.g., real zeros in periodic sequence.

Key Concepts

Twice Differentiable Functions

In differential calculus, a twice differentiable function is one that can be differentiated twice. This means it has a well-defined first and second derivative. Such functions are crucial when studying the rates at which changes occur, as well as how those rates themselves change. Consider the functions \( f_{1}(x) \) and \( f_{2}(x) \) in the given problem. They are specified as being twice differentiable, meaning we can reliably compute both their first and second derivatives, which in this context helps us understand the behavior of these functions over time.

Twice differentiable functions often display smoother behavior compared to functions that are only differentiable once. For example, they don't have sharp corners or discontinuities in their graphs. In applied mathematics, having twice differentiable functions allows for better modeling of physical phenomena because many physical laws are described by smooth, continuous curves.

Initial Value Problem

An initial value problem involves finding a function that satisfies a differential equation and specific initial conditions at a point. In this exercise, the functions \( f_1(x) \) and \( f_2(x) \) must meet the initial conditions \( f_1(0) = 2 \) and \( f_2(0) = 1 \). These conditions specify the values of the functions at \( x = 0 \), giving a particular solution to the differential equations \( f_{1}'(x) = f_2(x) \) and \( f_{2}'(x) = f_1(x) \).

Solving an initial value problem typically involves:

• Identifying the differential equation or system of equations.
• Applying the initial conditions to narrow down to a unique solution.
• Formulating the solution that satisfies both the equations and initial conditions.

Such problems are essential in fields like physics and engineering where initial states are known, and the objective is to understand how systems evolve over time.

Trigonometric Solutions

The differential equations provided, \( f_{1}'(x) = f_{2}(x) \) and \( f_{2}'(x) = f_{1}(x) \), strongly suggest that their solutions can be represented using trigonometric functions like sine and cosine. Trigonometric functions are periodic and smooth, ideal for modeling behaviors that repeat over regular intervals.

In solving such equations, we consider solutions of the form:

• \( f_1(x) = a\cos(x) + b\sin(x) \)
• \( f_2(x) = a\sin(x) - b\cos(x) \)

This choice stems from the properties of derivatives of sine and cosine, where the derivative of one leads naturally into another in a cyclic manner. Such trigonometric solutions simplify the problem significantly and can effectively describe oscillatory or wave-like systems.

System of Differential Equations

A system of differential equations involves multiple equations that define relationships between variables and their derivatives. In this exercise, the system is \( f_{1}'(x) = f_{2}(x) \) and \( f_{2}'(x) = f_{1}(x) \). Solving such systems often requires identifying patterns or using methods like substitution or elimination.

The solutions to these systems can describe complex phenomena where several factors interact. For instance, many mechanical systems, like linked pendulums, or electrical systems, such as circuits with inductors and capacitors, are modeled using systems of differential equations.

Techniques used to solve these systems include:

• Substitution to reduce multiple equations to a single equation.
• Matrix methods or eigenvectors in advanced applications.
• Numerical solutions when analytical solutions are challenging.

Understanding how to solve and apply these equations provides insight into multi-dimensional systems that change over time.