Final answer:
To solve the differential equation y⁴ + 20y" + 100y = 0 for the general solution y(x), we characteristically equate it to find the roots of the equation, which leads to a general solution that includes exponential, sine, and cosine functions with arbitrary constants.
Step-by-step explanation:
To find the general solution to the differential equation y⁴ + 20y" + 100y = 0, with x as the independent variable, we can treat this as a second-order linear homogeneous differential equation with constant coefficients. To solve it, we assume a solution of the form y = eˣₓ, where r is a constant that we need to determine. Plugging this into the differential equation gives us the characteristic equation:
r⁴ + 20r² + 100 = 0
This can be seen as a quadratic in terms of r², which leads to the next quadratic equation:
(r²)² + 20(r²) + 100 = 0
By solving this, we find two solutions for r², which then give us four solutions for r. The general solution will be a linear combination of these four solutions, normally involving exponential functions and possibly sine and cosine functions if the roots are complex. Given that the characteristic equation is (r² + 10)² = 0, we can see that r² = -10, which gives the solutions for r as ±10i. Consequently, the general solution will involve a combination of e⁰¹⁰ᵋ and e⁻¹⁰ᵋ terms, potentially multiplied by functions of x due to the multiplicity of the roots.
The complete general solution to the differential equation is:
y(x) = C₁e⁰¹⁰ᵋ(x)cos(10x) + C₂e⁰¹⁰ᵋ(x)sin(10x) + C₃xe⁰¹⁰ᵋ(x)cos(10x) + C₄xe⁰¹⁰ᵋ(x)sin(10x)
where C₁, C₂, C₃, and C₄ are arbitrary constants determined by the initial conditions.