Question

Homogeneous Equations with Constant Coefficients, Complex Roots of the Characteristic Equation, and Repeated Roots; Reduction of Order. (Remember the Discriminant?).

Find the most general solution (and explain your reasoning) to each of the following equations:
• y" – 54' + 6y = 0
• y'' + 4y = 0
• y'' – 6y' + 9y = 0

Answer 1

The question involves solving second-order linear differential equations. The solution includes finding the characteristic equation, determining the roots, and constructing the general solution based on the type of roots.

Step-by-step explanation:

To find the most general solution to the given second-order linear differential equations with constant coefficients, we use algebraic methods to solve for the characteristic equation and analyze the roots. Depending on the nature of the roots (real & distinct, real & repeated, or complex), we construct the general solution accordingly.

• For the equation y" – 54' + 6y = 0, the characteristic equation would be r^2 – 54r + 6 = 0. Solving this would give the roots necessary to write the general solution.
• The equation y'' + 4y = 0 corresponds to the characteristic equation r^2 + 4 = 0. Having complex roots, the general solution will involve sinusoidal functions.
• y'' – 6y' + 9y = 0 has the characteristic equation r^2 – 6r + 9 = 0, which yields repeated roots. The general solution therefore will use terms that account for the multiplicity of the root.

Each differential equation's solution involves the steps: finding the characteristic equation, solving for roots, constructing the general solution, checking for errors, and simplifying where possible.