Homogeneous Linear Third Order Differential Equation

Solve the homogeneous linear third order differential equation: $y''' - 7y'' - 8y' = 0$.

Solving higher order linear differential equations, such as this third order homogeneous one, involves techniques that extend the concepts used in second order equations. One key concept here is understanding the role of the characteristic equation in determining the form of the solution. For a third order equation, the characteristic equation is a cubic polynomial that will provide roots which indicate the general form of the solution. Depending on whether these roots are real and distinct, real and repeated, or complex, the solution will take different forms.

This problem also involves analyzing linear independence among solutions. Asserting linear independence is crucial for forming a complete solution set, ensuring that the principle of superposition can be effectively applied. If roots are repeated or complex, specific functions such as exponential functions combined with polynomials or sine and cosine functions will be incorporated to account for multiplicity or complex conjugates.

Mastering these problem-solving strategies not only helps in solving this specific type of differential equation, but also sets the foundation for tackling non-homogeneous equations and systems of equations where similar methodology is applied. Understanding the behavior and solutions of such equations is fundamental to modeling phenomena in engineering and physics, making these techniques highly applicable in real-world scenarios.