Unformatted text preview: MEC702 Wk 7 Lect 1 Nonhomogeneous Linear ODEs
We now turn from homogeneous to nonhomogeneous linear ODEs of nth order.
We write them in standard form as
y (n) + pn−1 (x)y (n−1) + · · · + p1 (x)y 0 + p0 (x)y = r(x) . Just like in Chapter 2, the solution of nonhomogeneous ODE is of the form
y = yh + yp where yh is the general solution for the homogeneous case and yp is the particular
solution found for the nonhomogeneous case.
• yh is found using λmethod.
• yp is found using Extended Method of Undetermined Coecients or Extended Method of Variation of Parameters. 1 Extended Method of Undetermined Coecients
This method still includes the same table used in Chapter 2:
Term in r(x)
γx ke
kx , (n = 0, 1, . . .)
k cos ωx
k sin ωx
keαx cos ωx
n Choice for yp (x)
Ceγx
Kn x + Kn−1 x
+ · · · + K1 x + K0 keαx sin ωx n n−1 K cos ωx + M sin ωx
eαx (K cos ωx + M sin ωx) The rules given in Chapter 2:
1. Basic Rule. If r(x) in ODE is one of the functions in the rst column
in the Table below, choose yp in the same line and determine its undetermined coecients by substituting yp and its derivatives into the ODE.
2. Modication Rule. If a term in your choice for yp happend to be a
solution of the homogeneous ODE corresponding to the nonhomogeneous
ODE, multiply this term by x (or x2 if this solution corresponds to a
double root of the characteristic equation of the homogeneous ODE.)
3. Sum Rule. If r(x) is a sum of functions in the rst column of the Table
below, choose for yp the sum of the functions in the corresponding lines
of the second column.
The rules for the extended method:
1. Extended Basic Rule: same as the one given in Chapter 2.
2. Extended Sum Rule: same as the one given in Chapter 2.
3. Extended Modication Rule: If the term y∗ in your choice of yp (x)
is a solution of the homogeneous case, then multiply this term by xk to
obtain
xk y∗, where k is smallest positive integer such that xk y∗ is not a solution of the
homogeneous case. Example. Solve the IVP
y 000 + 3y 00 + 3y 0 + y = 30e−x , y(0) = 3, y 0 (0) = −3, Solution. The particular solution of the IVP is
y = (3 − 25x2 )e−x + 5x3 e−x . Exercise. Solve the ODE
y 000 + 6y 00 + 12y 0 + 8y = 8x2 . 2 y 00 (0) = −47 . Extended Method of Variation of Parameters
The method of variation of Parameters given in Chapter 2 is given by
ˆ
yp (x) = −y1 y2 r
dx + y2
W ˆ y1 r
dx
W Now, extending this method to nth order nonhomogeneous ODEs with constant
coecients, we have
ˆ
n
X
Wk (x)
yp (x) =
yk (x)
r(x)dx
W (k)
k=1 or
ˆ
yp (x) = y1 (x)
ˆ
ˆ
W1 (x)
W2 (x)
Wn (x)
r(x)dx + y2 (x)
r(x)dx + · · · + yn (x)
r(x)dx
W (x)
W (x)
W (x) Example. Solve the EulerCauchy equation
x3 y 000 − 3x2 y 00 + 6xy 0 − 6y = x4 ln x, Solution. We have (x > 0) . yh = c1 x + c2 x2 + c3 x3 where the basis of solutions are
y1 = x, y2 = x2 , y3 = x3 . Then
W Then = 2x3 W1 = x4 W2 = −2x3 W3 = x2 .
11
1 4
.
yp = x ln x −
6
6 Exercise. Solve the ODE
y 000 + 6y 00 + 12y 0 + 8y = 8x2 . 3 ...
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 Winter '16
 Alveen Chand
 Trigraph, yp, 6y 00 + 12y 0 + 8y = 8x2

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