Roots of the characteristic equation - variation of

Question

1. e^it = cos t + i·sin t, t ∈ R.

2. ay′′ + by′ + cy = 0 (a ≠ 0)

its characteristic equation:

ar² + br + c = 0.

3. Method of Undetermined Coefficients: If in the equation

ay′′ + by′ + cy = g(t), a ≠ 0 e's t ∈ I

the right-hand side function g(t) has the form

g(t) = e^ut (A_n(t) cos(vt) + B_m(t) sin(vt)) ,

where A_n(t), B_m(t) are polynomials of degree n and m respectively, then the particular solution of the inhomogeneous equation has the form:

y_i,p = t^se^ut (P_k(t) cos(vt) + Q_k(t) sin(vt)),

where s is the multiplicity of the root u + i·v among the roots of the characteristic equation; further, P_k(t) and Q_k(t) are polynomials of degree k = max(n, m).

4. Variation of Parameters Method: Consider the inhomogeneous d.e.

y′′ + p(t)y′ + q(t)y = g(t) t ∈ I

and its homogeneous part Y′′ + p(t)Y′ + q(t)Y = 0. If the y₁, y₂ pair is a fundamental solution of the homogeneous d.e., then a particular solution of the inhomogeneous equation is looked for in the form y_i,p = C₁(t)·y₁(t) + C₂(t)·y₂(t), where for the derivatives of the unknown functions C₁(t), C₂(t) the following system of equations holds:

C′₁(t)y₁(t) + C'₂(t)y₂(t) = 0

C′₁(t)y₁(t) + C'₂(t)y₂(t) = g(t)

5. Special second order d.e.'s:

If y is missing, then substitute p(x) := y′(x).

If x is missing, then substitute q(y) := y′

6. The first order d.e. M (x, y)dx + N (x, y)dy = 0 is exact, if

∂M/∂y = ∂N/∂x

To solve the d.e., a function F : R² → R has to be found such that gradF = (M, N). Then the solution of the d.e. is:

F (x, y) = Const.