Find a finite set of real functions that spans the solution...

Find a finite set of real functions that spans the solution space to the following homogeneous ordinary differential equations: (a) $u^{\prime}-2 u=0$, (b) $u^{\prime \prime}+4 u=0$, (c) $u^{\prime \prime}-3 u^{\prime}=0$, (d) $u^{\prime \prime}+u^{\prime}+u=0$, (e) $u^{\prime \prime \prime}-5 u^{\prime \prime}=0$, (f) $u^{(4)}+u=0$.

Find a finite set of real functions that spans the solution space to the following homogeneous ordinary differential equations: (a) $u^{\prime}-2 u=0$, (b) $u^{\prime \prime}+4 u=0$, (c) $u^{\prime \prime}-3 u^{\prime}=0$,
(d) $u^{\prime \prime}+u^{\prime}+u=0$,
(e) $u^{\prime \prime \prime}-5 u^{\prime \prime}=0$,
(f) $u^{(4)}+u=0$.

Key Concepts

Multiplicity of Roots and Generalized Eigenfunctions

In cases where the characteristic equation has repeated roots, the general solution must include not only the exponential solution but also terms multiplied by powers of the independent variable. This accounts for the multiplicity of the eigenvalue and ensures that the resulting set of solutions is linearly independent, thereby forming a complete basis for the solution space.

Basis of the Solution Space

For any n-th order linear homogeneous differential equation, the solution space is n-dimensional, meaning it has a basis consisting of n linearly independent functions. Finding such a basis is crucial because it guarantees that every solution can be written as a linear combination of these basis functions, which simplifies the analysis and application of the general solution in various problems.

Complex Roots and Trigonometric Functions in ODE Solutions

When the characteristic equation yields complex roots, the corresponding solutions involve complex exponentials, which can be rewritten in terms of sine and cosine functions using Euler’s formula. This is essential to provide a real-valued fundamental set of solutions, which is often needed in physical problems and real-world applications.

Characteristic Equation

The characteristic equation is obtained by substituting an exponential trial solution into a constant coefficient linear differential equation, which converts the differential equation into an algebraic equation, typically a polynomial. The roots of this polynomial determine the form of the solutions to the differential equation, making this step essential in the systematic solving of many differential equations.

Exponential Function Solutions

Exponential functions play a central role in solving constant coefficient differential equations because they are eigenfunctions of differentiation. When an exponential function is substituted into a differential equation, its derivatives remain proportional to the function itself. This makes them natural candidates for formulating the general solution, particularly when the characteristic equation yields real roots.

Constant Coefficient Equations

Differential equations with constant coefficients have the advantage that the coefficients of the function and its derivatives do not depend on the independent variable. This property allows one to look for solutions in the form of exponential functions, leading directly to a characteristic polynomial. The constant coefficient assumption simplifies the analysis and solution process, making it a staple concept in the study of ordinary differential equations.

Linear Homogeneous Differential Equations

A linear homogeneous differential equation is an equation in which the unknown function and its derivatives appear linearly and without a forcing (nonhomogeneous) term. The general solution to such an equation is composed of a linear combination of independent solutions. This concept is fundamental because it guarantees that if you have a set of linearly independent solutions, then any solution can be expressed as a sum of these multiplied by arbitrary constants, forming a vector space.

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