Question

The non-homogeneous problem

y''+16y=48tan(4x)
The general solution of the homogeneous problem (called the complementary solution,
y=ay₁+by₂ )is given in terms of a pair of linearly independent solutions,y₁, y₂. Here a and b are
arbitrary constants.
Find a fundamental set for y''+16y=0

Answer 1

The fundamental set for the homogeneous differential equation y''+16y=0 is composed of y1 = cos(4x) and y2 = sin(4x), which are both solutions derived from the characteristic equation with imaginary roots.

Step-by-step explanation:

To find a fundamental set of solutions for the homogeneous differential equation y''+16y=0, we can make an educated guess that the solutions might take the form of trigonometric functions, because the differential equation resembles the form of a classical harmonic oscillator equation. The characteristic equation is obtained by replacing y with ert, where r is some constant, resulting in the equation r2+16=0. Solving for r, we get r = ±4i, which are purely imaginary roots.

This leads us to the general solution for the complementary (homogeneous) equation, which can be written in terms of sine and cosine functions because of the imaginary roots. Therefore, y1 = cos(4x) and y2 = sin(4x) are the linearly independent solutions to this equation. Hence, we have our fundamental set of y1 and y2, representing the solution space of the differential equation.