Mae675 Fall 2020 Homework 2 Note

Mae675 Fall 2020 Homework 2note This Homework Is Given Wednesday Se

This homework assignment involves solving differential equations, analyzing their properties, and applying special functions like Chebyshev polynomials. The tasks include finding fundamental solutions to second-order linear equations, exploring the concept of exact and self-adjoint differential equations, applying the method of variation of parameters, deriving the convolution integral solution, and investigating Chebyshev differential equations including their polynomial solutions.

Paper For Above instruction

The assignment encompasses a comprehensive study of second-order differential equations, their classifications, solutions, and special functions. It begins with analyzing a mass-spring-damper system described by a second-order linear homogeneous differential equation with initial conditions. The goal is to find a fundamental set of solutions by reduction of order and then solve the initial value problem explicitly.

Next, the focus shifts to the concept of exact differential equations. The problem requires demonstrating the necessary condition for an equation to be exact, deriving the adjoint (integrating factor) equation, and investigating whether the Airy equation is self-adjoint. This involves understanding the structure of differential operators and their symmetries, especially for special equations like the Airy equation, which arises in various physical contexts.

The third problem explores the method of variation of parameters for solving nonhomogeneous linear second-order differential equations. It involves deriving the general solution using convolution integrals and applying initial conditions to obtain a specific solution. This enhances understanding of how external forcing functions influence the system's response over time, especially when viewed as a signal-processing operation.

Finally, the assignment examines Chebyshev equations, solutions in terms of power series near |x|

Complete Solution to the Assignment

Problem 1: Homogeneous System, Reduction of Order, and Initial Conditions

The differential equation is:

with initial conditions y(0) = 0.3, y'(0) = -0.15.

The characteristic equation is:

which factors as:

The repeated root is r = -0.6, indicating a solution of the form:

Applying initial conditions:

and at x=0:

Substituting A = 0.3:

Final solution:

Problem 2: Exact Equations and Self-Adjointness

Given the general form:

a. The equation is exact if it can be written as:

The common criteria provide the necessary condition:

(which is also sufficient for exactness). This involves differentiation of coefficient functions.

b. To find the integrating factor μ(x), the differential equation governing μ(x) is:

This is an adjoint equation derived from the integrating factor technique, allowing the differential equation to be made exact when multiplied by μ(x).

c. For the Airy equation:

as it can be rewritten as:

Q(x) = 0,

R(x) = -x.

The adjoint equation derived from the general form:

which simplifies to:

or:

d. For the equation to be self-adjoint, the necessary condition is P'(x) = Q(x). In the Airy equation, P'(x)=0 and Q(x)=0, satisfying this condition. Hence, the Airy equation is self-adjoint.

Problem 3: Variation of Parameters and Convolution Solution

a. The homogeneous solution of:

is:

Using variation of parameters, the particular solution with nonhomogeneous term g(x) is:

which, considering initial conditions, simplifies to:

b. To satisfy initial conditions y(x_0) = 0 and y'(x_0) = 0:

leading to the general solution:

c. Incorporating initial conditions y(0) = y₀, y'(0) = y'₀:

This combines the homogeneous solution components with the integral of the forcing function, illustrating the superposition principle in linear differential equations.

Problem 4: Chebyshev Differential Equation and Polynomial Solutions

a. The Chebyshev differential equation:

Finding solutions in power series for |x|

b. Numerical solutions using Maple's dsolve with series expansion or analogous commands in MATLAB or Mathematica confirm the power series results. Graphs of these solutions reveal how polynomial solutions approximate the function within the interval, with convergence improving as more terms are included.

c. When α is a non-negative integer n, solutions become finite degree polynomials:

are explicitly given by:

T_1(x)=x,

T_2(x)=2x^2 - 1,

T_3(x)=4x^3 - 3x.

These polynomials satisfy the orthogonality conditions and are useful in spectral methods and approximation problems.

d. Explicit polynomial solutions for degrees n=0,1,2,3:

T_1(x) = x,

T_2(x) = 2x^2 - 1,

T_3(x) = 4x^3 - 3x,

which are derived directly by solving the differential equation or recognizing their standard forms.

Conclusion

This comprehensive exploration of second-order differential equations illustrates fundamental methods such as reduction of order, the concept of exact and self-adjoint equations, the application of variation of parameters, and polynomial solutions of special functions. The analyses of the Airy and Chebyshev equations deepen understanding of their properties and applications in physics, engineering, and numerical analysis, highlighting the interconnectedness of differential equations, linear algebra, and orthogonal polynomials.

References

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• Arfken, G. B., Weber, H. J., & Harris, F. E. (2013). Mathematical Methods for Physicists. Academic Press.
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