Equation with a right-hand side: particular solution

Structure of the general solution

For ay'' + by' + c*y = f(x), the general solution is given by: y = y_h + y_p where y_h is the general solution of the associated homogeneous equation (as seen previously) and y_p is any particular solution of the complete equation.

Solving by Identification

We choose a form for y_p that is suitable for f(x):

Type of f(x)	Form of y_p to try
Polynomial of degree n	Polynomial of degree n (or n+1 if 0 is a root of the characteristic equation)
K * exp(m*x)	Aexp(mx) (or Axexp(mx) if m is a simple root, Ax²exp(mx) if it is a double root)
Kcos(wx) or Ksin(wx)	Acos(wx) + Bsin(wx)

Example

Solve y'' - 3y' + 2y = e^(3x). Homogeneous (as seen previously): y_h = C1exp(x) + C2exp(2x). As 3 is not a root of the characteristic equation, we try y_p = Aexp(3x). By differentiating and rearranging: (9A - 9A + 2A)exp(3x) = e^(3x) → 2A = 1 → A = 1/2 General solution: y(x) = C1exp(x) + C2exp(2x) + (1/2)*exp(3x)

Common pitfall: the resonance phenomenon

If the term exp(mx) on the right-hand side has m equal to a root of the characteristic equation, trying Aexp(m*x) yields 0 = f(x), which is absurd. We must therefore multiply by x (or x² if the root is double): this is the resonance phenomenon, common in physics for oscillators driven at their natural frequency.

Determination of the constants

Once y = y_h + y_p has been found, the constants C1 and C2 are determined by two initial conditions (y(x0) and y'(x0)), as the space of homogeneous solutions is two-dimensional.