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< is given by
y = [Sigma]c_m(D + a_m)^-1 [psi](x) = [Sigma]c_m(D + a_m)^-1 e^-a m^x e^a m^x [psi](x) =
[Sigma]c_m e^-a m^x D^-1 (e^a m^x [psi](x)) = [Sigma]c_m e^-a m^x [int] e^a m^x [psi](x)dx.
The particular integral is thus expressed as a sum of n integrals.
A linear differential equation of which the left side has the form
d^ny d^n-1 y dy
x^n ---- + P1x^n-1 ------- + ... + P_n-1 x-- + P_n y,
dx^n dx^n-1 dx
where P1, ... Pn are constants, can be reduced to the case considered
above. Writing x = e^t we have the identity
d^mu
x^m ---- = [t]([t] - 1)([t] - 2) ... ([t] - m + 1)u, where [t] = d/dt.
dx^m
When the linear differential equation, which we take to be of the
second order, has variable coefficients, though there is no general
rule for obtaining a solution in finite terms, there are some results
which it is of advantage to have in mind. We have seen that if one
solution of the equation obtained by putting the right side zero, say
y1, be known, the equation can be solved. If y2 be another solution of
d^2y dy
---- + P-- + Qy = 0,
dx^2 dx
there being no relation of the form my1 + ny2 = k, where m, n, k are
constants, it is easy to see that
d/dx(y1'y2 - y1y2') = P(y1'y2 - y1y2'),
so that we have
y1'y2 - y1y2' = A exp.([int] Pdx),
where A is a suitably chosen constant, and exp. z denotes e^z. In
terms of the two solutions y1, y2 of the differential equation having
zero on the right side, the general solution of the equation with R =
[phi](x) on the right side can at once be verified to be Ay1 + By2 +
y1u - y2v, where u, v respectively denote the integrals
_ _
/ /
u = |y2[phi](x)(y1'y2 - y2'y1)^-1 dx, v = |y1[phi](x)(y1'y2 - y2'y1)^-1 dx.
_/ _/
The equation
d^2y dy
---- + P-- + Qy = 0,
dx^2 dx
by writing y = v exp. (-1/2 [int] Pdx), is at once seen to be reduced to
d^2v/dx^2 + 1v = 0, where 1 = Q - 1/2dP/dx - 1/4P^2. If [eta] = - 1/v dv/dx,
the equation d^2v/dx^2 + 1v = 0 becomes d[eta]/dx = 1 + [eta]^2, a
non-linear equation of the first order.
More generally the equation
d[e]]>
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