Introduction To Second Order Linear NonHomogeneous Differential Equations

The nonhomogeneous differential equation of this type has the form

ay′′+by′+cy=f(x),

where a, b and c are constant numbers.
For each equation we can write the related homogeneous or complementary equation:

ay′′+by′+cy=0

Theorem.

The general solution of a nonhomogeneous equation is the sum of the general solution yₕ(x) of the related homogeneous equation and a particular solution yₚ(x) of the nonhomogeneous equation:

y(x)=yₕ(x)+yₚ(x).

Method of Undetermined Coefficients

The right side f(x) of a nonhomogeneous differential equation is often an exponential, polynomial or trigonometric function or a combination of these functions. In this case, it’s more convenient to look for a solution of such an equation using the method of undetermined coefficients.

The given method works only for a restricted class of functions in the right side, such as

f(x) =Pₙ(x)eᵃˣ;

f(x) =[Pₙ(x)cos(βx) + Qₘ(x)sin(βx)]eᵃˣ,

where Pₙ(x) and Qₘ(x) are polynomials of degree n and m, respectively.

In both cases, a choice for the particular solution should match the structure of the right side of the nonhomogeneous equation.

In case 1, if the power α of the exponential function coincides with a root of the auxiliary characteristic equation, the particular solution will contain the additional factor xˢ, where s is the order of the root α in the characteristic equation.

In case 2, if the number α+βi coincides with a root of the characteristic equation, the expected expression for the particular solution should be multiplied by the additional factor x.

The unknown coefficients can be determined by substitution of the expected type of the particular solution into the original nonhomogeneous differential equation.

Видео Introduction To Second Order Linear NonHomogeneous Differential Equations канала Tambuwal Maths Class