Question

Using the perturbation theory of Section \(10-2\), calculate \(\alpha(\beta)\) for the periodic solution of the Mathieu equation which reduces to \(\sin x\) as \(\beta \rightarrow 0\). Retain terms up to and including \(\beta^{4}\).

Short answer

To the fourth order, \[ \alpha(\beta) = 1 + \alpha_2\beta^2 + \alpha_4\beta^4 \].

Step-by-step solution

Understand the Mathieu Equation

The standard form of the Mathieu equation is given by \[ \frac{d^2y}{dx^2} + ( \alpha - 2\beta \cos(2x) ) y = 0 \] where \[ \alpha \] and \[ \beta \] are parameters.

Express the Solution in Perturbation Series

We express \[ y(x) \] in a perturbation series as \[ y(x) = y_0(x) + \beta y_1(x) + \beta^2 y_2(x) + \beta^3 y_3(x) + \beta^4 y_4(x) + \ldots \]

Assume the Form of the Solution

We assume \[ y_0(x) = A \sin(x) \] since it reduces to \[ \sin(x) \] as \[ \beta \rightarrow 0 \].

Substitute into the Differential Equation

Substitute the assumed series \[ y(x) \] into the original Mathieu equation and collect terms of \[ \beta^0, \beta^1, \beta^2, \beta^3, \beta^4 \]. This yields a system of differential equations for \[ y_0, y_1, y_2, y_3, y_4 \].

Solve for the Zeroth Order

To zeroth order in \[ \beta \] we have \[ \frac{d^2 y_0}{dx^2} + \alpha y_0 = 0 \]. Given \[ y_0 = \sin(x) \], we obtain \[ \alpha = 1 \].

Solve for the First Order

To first order in \[ \beta \], we have \[ \frac{d^2 y_1}{dx^2} + \alpha y_1 = 2\cos(2x)y_0 \]. Substitute \[ y_0 = \sin(x) \], and solve for \[ y_1 \] using appropriate methods.

Higher-Order Corrections

Follow similar steps for higher-order terms (up to \[ \beta^4 \]) to determine \[ y_2, y_3, y_4 \] and correct \[ \alpha \].

Assemble the Full Solution

Combine all terms to express the periodic solution to the desired order of \[ \beta \]. The final expression for \[ \alpha(\beta) \] can be written as \[ \alpha(\beta) = 1 + \alpha_2\beta^2 + \alpha_4\beta^4 \]. These coefficients can be deduced from specific calculations in previous steps.

Key concepts

Mathieu equation

The Mathieu equation is a type of differential equation that appears in various fields such as physics and engineering. It is expressed as \[ \frac{d^2y}{dx^2} + ( \alpha - 2\beta \cos(2x) ) y = 0 \]where \( \alpha \) and \( \beta \) are parameters. This equation describes systems where parameters vary periodically, like vibrating membranes or electron behavior in crystals. The challenge with the Mathieu equation lies in its dependency on the cosine function, making its solutions periodic—yet non-trivial to find. By understanding the parameters and structure of this equation, we can apply perturbation methods to find approximate solutions.

perturbation theory

Perturbation theory is a mathematical approach where we start with a known solution of a simpler problem and incrementally build up to the solution of a more complex problem by introducing a small 'perturbing' parameter. For the Mathieu equation, \( \beta \) acts as this small perturbing parameter. The solution \( y(x) \) is expressed as a series:\[ y(x) = y_0(x) + \beta y_1(x) + \beta^2 y_2(x) + \beta^3 y_3(x) + \beta^4 y_4(x) + \ldots \]Step-by-step, we solve each term (\( y_0, y_1, y_2, ... \) ). This method is particularly useful when the parameter \( \beta \) is small, allowing simplified approximate solutions.

differential equations

Differential equations involve functions and their derivatives, describing how a particular quantity changes over time. The solution to a differential equation provides a function or a set of functions that satisfy the given equation. In the case of the Mathieu equation, we deal with second-order differential equations. Solving these involves finding functions \( y(x) \) that satisfy the equation under given conditions. By expressing \( y(x) \) as a series and substituting back into the original equation, we derive a system of simpler differential equations for each term in the series.

periodic solutions

Periodic solutions are functions that repeat their values in regular intervals. For the Mathieu equation, we seek solutions that repeat every \( 2\pi \) like sine and cosine functions. Initially, we assume \( y_0(x) = A \sin(x) \) as our primary solution since it matches the periodic behavior of the Mathieu equation for \( \beta=0 \) . Higher-order terms refine this basic solution to account for the effects of the parameter \( \beta \) . The primary solution ensures that as \( \beta \rightarrow 0 \) , the solution will still behave like \( \sin(x). \) Periodic solutions are crucial in real-world applications where repetition over time is observed, such as in mechanical vibrations and wave phenomena.

higher-order terms

Higher-order terms in the perturbation series allow for more precise solutions by including additional corrections. For the Mathieu equation, after determining the first-order correction term \( y_1 \) , we continue to higher orders like \( y_2 \) , \( y_3 \) , etc. These corrections help refine the parameter \( \alpha \) with each order, leading to an expression like \( \alpha(\beta) = 1 + \alpha_2\beta^2 + \alpha_4\beta^4 \) . The coefficients \( \alpha_2 \) and \( \alpha_4 \) are calculated based on the perturbation series. By including higher-order terms, our solution approximations become more accurate, reflecting the true behavior of the system more closely.