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Section C.4 Linear Homogeneous Constant Coefficients

N-th Derivative of erx.

erxr,
chain  rule  [erx]=rerx,[erx]=r2erx,,[erx](n)=rnerx.

2nd Order LHCC Complex Case 3.

The following explains how Case 3 (complex) comes directly from Case 1 (real & ).
Since r1 and r2 are complex, they can be written as
r1=α+βiandr2=αβi.
Substituting these into (30), using Euler’s Formula, the even property of cosine and odd property of sine, we can rewrite the general solution as
y =C1e(α+βi)x+C2e(αβi)x=C1eαxeiβx+C2eαxeiβx=eαx[C1eiβx+C2eiβx]=eαx[C1(cos(βx)+isin(βx))+C2(cos(βx)+isin(βx))]=eαx[C1cos(βx)+iC1sin(βx)+C2cos(βx)+iC2sin(βx)]=eαx[C1cos(βx)+iC1sin(βx)+C2cos(βx)iC2sin(βx)]=eαx[(C1+C2C1)cos(βx)+i(C1C2)C2sin(βx)]=eαx(C1cos(βx)+C2sin(βx)).

Like Terms.

Terms are called like terms if they have identical variable parts. That is, they differ only by a coefficient. Like terms can be combined via addition and subtraction. For example, the x2 and e3x terms below are pairs of like terms, which can be combined as follows:
3x2+5e3x2+7x24e3x10x2+e3x2.

Polynomial Factoring Calculator.

Figure C.10. Type in any polynomial of r. Use “” for multiplication. If the polynomial has rational roots, it will show the factors.
To find the characteristic equation, we substitute y=erx into the n-th order LHCC equation and solve for r. Doing this gives us
an [erx](n)++a2 [erx]+a1 [erx]+a0 erx= 0an rnerx++a2 r2erx+a1 rerx+a0 erx= 0(an rn++a2 r2+a1 r+a0)erx= 0.
Since erx is never zero, we must have
an rn++a2 r2+a1 r+a0=0
which is an n-th order polynomial in r.
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