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The method for solving nonhomogeneous second-order differential equations by separating variables and finding complementary functions yc and particular solutions yp. It covers two cases: when f(x) is a constant, exponential, or a combination of cosine and sine functions. Examples and step-by-step solutions.
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Nonhomogeneous Second-Order Differential Equations To solve ay′′^ + by′^ + cy = f (x) we first consider the solution of the form y = yc + yp where yc solves the differential equaiton ay′′^ + by′^ + cy = 0 and yp solves the differential equation ay′′^ + by′^ + cy = f (x). Since the derivative of the sum equals the sum of the derivatives, we will have a final solution of 0 + f (x) which gives f (x). Therefore, we follow a pattern for yp to yield f (x) when each of y, y′^ and y′′^ are substi- tuted into the above equation.
(a) f (x) = x^2 e^3 x. Set yp = (Ax^2 + Bx + C)e^3 x. (b) f (x) = x^3 − 4 x + 1. Set yp = Ax^3 + Bx^2 + Cx + D. (c) f (x) = 6e−^2 x. Set yp = Ae−^2 x.
Then we substitute yp, y p′, and y′′ p and solve for the coefficients.
(a) f (x) = 2 cos (3x). Set yp = A cos (3x) + B sin (3x) (b) f (x) = x cos (x). Set yp = (Ax + B) cos (x) + (Cx + D) sin (x) (c) f (x) = ex^ sin (2x). Set yp = Aex^ sin (2x) + Bex^ cos (2x)
If f (x) is a sum of terms, like f (x) = x^2 + e−x^ + cos (x), do it as separate problems solving for yp 1 = Ax^2 + Bx + C, yp 2 = Ae−x^ and yp 3 = A cos (x) + B sin (x).
Examples
y = yc + yp
r^2 + 2r + 1 = 0. r = −1. Therefore yc = Me−x^ + Nxe−x^ (hold until end). Now to determine yp, consider the form: yp = A cos (2x) + B sin (2x). Then differentiating gives y p′ = − 2 A sin (2x) + 2B cos (2x) y p′′ = − 4 A cos (2x) − 4 B sin (2x). Substituting gives: − 4 A cos (2x) − 4 B sin (2x) − 4 A sin (2x) + 4B cos (2x) + A cos (2x) + B sin (2x) = 3 cos (2x) So that: − 4 A + 4B + A = 3 and − 4 B − 4 A + B = 0 The system of equations gives A = − 259 and B = 1225. NOT DONE: We now have
y = Me−x^ + Nxe−x^ −
cos (2x) +
sin (2x)
Substituting the initial conditions gives M = 259 and N = 1025. DONE
y =
e−x^ +
xe−x^ −
cos (2x) +
sin (2x)