Problem 43 - Advanced Engineering Mathematics

Question

\(y=c_{1} \cos 3 x+c_{2} \sin 3 x\) is a two-parameter family of solutions of the second-order DE \(y^{\prime \prime}+9 y=0\). If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. $$ y(0)=0, y(\pi)=4 $$

Step-by-Step Solution

Answer

No solution exists under the given boundary conditions; they are inconsistent.

Step 1: Verify the General Solution

The general solution of the differential equation \( y'' + 9y = 0 \) is given as \( y = c_1 \cos 3x + c_2 \sin 3x \). To verify, differentiate twice: First derivative: \( y' = -3c_1 \sin 3x + 3c_2 \cos 3x \).Second derivative: \( y'' = -9c_1 \cos 3x - 9c_2 \sin 3x \).Substitute back in the DE: \( -9c_1 \cos 3x - 9c_2 \sin 3x + 9(c_1 \cos 3x + c_2 \sin 3x) = 0 \), which simplifies to \( 0 = 0 \). This confirms that \( y = c_1 \cos 3x + c_2 \sin 3x \) is indeed a solution.

Step 2: Apply the First Boundary Condition

Apply the first boundary condition \( y(0) = 0 \): Set \( y = c_1 \cos 3(0) + c_2 \sin 3(0) \), which simplifies to \( y = c_1 \). So, \( c_1 = 0 \) from \( y(0) = 0 \). Thus, the reduced general solution is \( y = c_2 \sin 3x \).

Step 3: Apply the Second Boundary Condition

Now, apply the second boundary condition \( y(\pi) = 4 \): Set \( y = c_2 \sin 3(\pi) \). Since \( \sin 3\pi = 0 \), \( y(\pi) = c_2 \cdot 0 = 0 \), which cannot equal 4.Therefore, no solution exists under these boundary conditions because \( c_2 \sin 3x \) cannot be non-zero at \( x = \pi \).

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They express a relationship between a function and the rate of change of the function. Differential equations play a key role in fields like physics, engineering, biology, and economics, as they are used to model real-world situations where quantities change.

• Order: The order of a differential equation is the highest derivative present in the equation. For example, if the highest derivative is the second derivative, it's called a second-order differential equation.
• Linearity: Linear differential equations are those in which the dependent variable and its derivatives appear linearly (no products or powers of derivatives). Conversely, if terms are quadratic or involve products of derivatives, the equation is nonlinear.

Differential equations can be complex, but they are soluble in various ways, often involving specific initial or boundary conditions that permit the calculation of particular solutions from a general solution.

Second-order Differential Equation

Second-order differential equations involve derivatives up to the second degree. They take the form \( a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x) \). These equations can describe a variety of phenomena including oscillations, waves, and other dynamic systems.

• Solving Second-order Equations: Solutions to these equations often require methods such as the characteristic equation, variation of parameters, or method of undetermined coefficients. For homogeneous equations like \( y'' + 9y = 0 \), solutions typically involve exponential or trigonometric functions.
• Trigonometric Solutions: In cases like our example, trigonometric functions arise because of the nature of the equation. The solution \( y = c_1 \cos(3x) + c_2 \sin(3x) \) corresponds to the characteristic equation having imaginary roots.

These equations are powerful tools in modeling behaviors like simple harmonic motion, where the system exhibits periodicity and stability over time.

Trigonometric Functions

Trigonometric functions such as sine and cosine are commonly used in solutions to differential equations, especially those involving periodic phenomena.

• Fundamentals: The sine and cosine functions are wave-like functions that have applications in modeling oscillatory motions. They are defined for all real numbers and have a range between -1 and 1.
• Usage in Differential Equations: In solving differential equations, these functions help express solutions for systems with cyclic behavior. For example, in the equation \( y'' + 9y = 0 \), solutions feature \( \cos(3x) \) and \( \sin(3x) \) terms due to the form of the equation.

While these functions simplify understanding of periodic systems, they also help meet boundary conditions by adjusting coefficients. However, at certain boundaries, they may not satisfy required conditions, which can imply nonexistence of solutions given the constraints.