Ch 08/Second Order Linear DEs with Constant Coefficients/H.K. Dass 3.18-3.25
Second Order Linear DEs with Constant Coefficients
For ay'' + by' + cy = R(x), the complete solution is y = C.F. + P.I. The C.F. comes from the auxiliary equation; the P.I. depends on the form of R(x).
IThe Complementary Function
General form: . Solution:
For C.F., set , assume . Get the Auxiliary Equation: .
Case I -- Distinct real roots $m_1, m_2$: C.F.
Case II -- Repeated roots $m, m$: C.F.
Case III -- Complex roots $\alpha \pm i\beta$: C.F.
Note
The A.E. is just a quadratic. Use the quadratic formula if factoring is difficult.
IIParticular Integrals
Write where . Then P.I. .
Type 1: $R = e^{ax}$ P.I. . If , multiply by .
Type 2: $R = \sin ax$ or $\cos ax$ Replace by . If 0 results, multiply by .
Type 3: $R = x^n$ Expand by binomial series.
Type 4: $R = e^{ax}\phi(x)$ P.I. (exponential shift).
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