Differential Equations Workbook For Dummies®

5.2. Getting the General Solution When g(x) Is a Polynomial

Sometimes g(x) acts as a polynomial, like in the case of g(x) = axⁿ + bx^{n − 1} + cx^{n − 2} (where a, b, and c are all constants). You need to know how to handle such situations so you can find the general solution to the nonhomogeneous second order differential equation in question.

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Here's how to solve a differential equation of this form, ay″ + by′ + cy = g(x), where a, b, and c are constants and g(x) is a polynomial of order n:

1. If g(x) is a polynomial, you can assume the particular solution is of the same form, using coefficients whose values have yet to be determined.

   y_p = A_nxⁿ + A_{n − 1} x^{n − 1} + A_{n − 2}x^{n − 2} + ...+ A₁x + A₀

2. If g(x) is the sum of terms, g₁(x), g₂(x), g₃(x) and so on, then you can break the problem into various subproblems.

   ay″ + by′ + cy = g₁(x)

   ay″ + by′ + cy = g₂(x)

   ay″ + by′ + cy = g₃(x)

   The particular solution is the sum of the solutions of these subproblems.

3. Substitute y_p into the differential equation and solve for the undetermined coefficients.

4. Find the general solution, y_h = c₁y₁ + c₂y₂, of the associated homogeneous differential equation.

5. The general solution of the nonhomogeneous differential equation is the sum of y_h and y_p.

6. Use the initial conditions to solve for c₁ and c₂, if the problem calls for that.

Take a look at the following example. Then, if you're game, try the following practice problems that ask you to solve for the general solution when g(x) is a polynomial.

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EXAMPLE

Q. Find the ...