Explain why the functions 1, cosx, sinx span the solution space to the third order ordinary diff

step 1 In this problem, we have y double dash plus 2 y dash plus 4 y is equals to 5 sin t. Now, we are

Explain why the functions 1, cosx, sinx span the solution...

Key Concepts

Order of a Linear Differential Equation

A linear homogeneous ordinary differential equation (ODE) of order n has a solution space that is an n-dimensional vector space. In this context, since the given ODE is third order, any set of three linearly independent solutions forms a basis, thereby spanning the entire solution space.

Characteristic Equation

To solve a constant coefficient linear ODE, one typically assumes a solution of the form u(x) = e^(rx). Substituting this into the differential equation yields the characteristic equation, which in this case is r^3 + r = 0. The roots of this polynomial determine the structure of the general solution.

Complex Roots and Real-Valued Solutions

The characteristic equation r^3 + r = 0 factors as r(r^2 + 1) = 0, yielding one real root (r = 0) and a pair of complex conjugate roots (r = i and r = -i). Complex roots of the form ±i lead to real-valued solutions typically expressed using sine and cosine functions through Euler's formula, thereby giving rise to cos x and sin x as part of the general solution.

Span and Basis of the Solution Space

The functions 1, cos x, and sin x are linearly independent and provide a basis for the three-dimensional solution space of the given third order ODE. This means any solution to the ODE can be expressed as a linear combination of these three functions, and they therefore span the entire solution space.

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