Nonhomogeneous Linear Differential
EquationsAP CALCULUS BC
Nonhomogeneous Differential EquationsRecall that second order linear differential equations with constant coefficients have the form:
Now we will solve equations where G(x) ≠ 0, which are non-homogeneous differential equations.
( )ay by cy G x
Complementary EquationThe related homogeneous equation is called the complementary equation, and it is part of the general solution to the nonhomogeneous equation.
0ay by cy
General SolutionThe general solution to a nonhomogenous differential equation is
y(x) = yp(x) + yc(x)
where yp is a particular solution to the nonhomogenous equation, and yc
is the general solution to the complementary equation.
Proof for Grant
yc(x) = y(x) – yp(x)( ) ( ) ( )p p pa y y b y y c y y
p p pay ay by by cy cy
( ) ( )p p pay by cy ay by cy ( ) ( ) 0g x g x
Method of Undetermined Coefficients
There are two methods for solving nonhomogeneous equations:Method of Undetermined CoefficientsVariation of Parameters
First, we will learn about the Method of Undetermined Coefficients to solve the equation ayʹʹ + byʹ + cy = G(x) when G(x) is a polynomial.
Since G(x) is a polynomial, yp is also a polynomial of the same degree as G. Therefore, we substitute yp = a polynomial (of the same degree as G) into the equation and determine the coefficients.
Example 1 Solve the equation yʹʹ + yʹ – 2y = x2.
The auxiliary equation is r2 + r – 2 = 0
Factor (r – 1)(r + 2) = 0
r = 1, r = –2
Solution of complementary equation is yc = c1ex + c2e–2x
Since G(x) = x2, we want a particular solution where
yp(x) = Ax2 + Bx + C
So ypʹ(x) = 2Ax + B and ypʹʹ(x) = 2A
Example 1 (continued) Substitute these into the differential equation (2A) + (2Ax + B) – 2(Ax2 + Bx + C) = x2
= – 2Ax2 + (2A – 2B)x + (2A + B – 2C) = 1x2 + 0x + 0
–2A = 1
2A – 2B = 0
1
2A
12 2 0
2B
1
2B
2 2 0A B C 1 1
2 2 02 2
C
3
4C
Example 1 (FINAL) Therefore, our particular solution is
So our general solution is:
y = yc + yp = c1ex + c2e–2x
21 1 3
2 2 4py x x
21 1 3
2 2 4x x
Example 2 Solve yʹʹ + 4y = e3x
When G(x) is of the form ekx, we use yp = Aekx because the derivatives of ekx are constant multiples of ekx and work out nicely.
Complementary equation is r2 + 4 = 0
Therefore, yc = c1 cos 2x + c2 sin 2x
Solve for yp:
y = Ae3x yʹ = 3Ae3x yʹʹ = 9Ae3x
Substitute
2r i
3 3 39 4x x xAe Ae e
Example 2 (continued)
13A = 1
A = 1/13
General solution is
3 3 39 4x x xAe Ae e 3 313 x xAe e
31
13x
py e
31 2
1cos2 sin 2
13xy c x c x e
Example 3 Solve the equation yʹʹ + yʹ – 2y = sin x.
When G(x) is of the form C sin kx or C cos kx, we use yp = A cos kx + B sin kx
Complementary equation is r2 + r – 2 = 0 r = –2, 1
yp = A cos x + B sin x
ypʹ = –A sin x + B cos x
ypʹʹ = –A cos x – B sin x
( 2)( 1) 0r r 2
1 2x x
cy c e c e
Example 3 (continued) Substitute back into original equation:
(–A cos x – B sin x) + (–A sin x + B cos x) – 2(A cos x + B sin x) = sin x
(–3A + B) cos x + (–A – 3B) sin x = sin x
Therefore, –3A + B = 0 and –A – 3B = 1
Solve as a system
General solution is
1 3
10 10A B
1 3cos sin
10 10py x x
21 2
1 3cos sin
10 10x xy c e c e x x
Review and More Rules for Method of Undetermined Coefficients
Form is ayʹʹ + byʹ + cy = G(x)
1. If G(x) is a polynomial, use yp = Axn + Bxn–1 + … + C.
2. If G(x) = Cekx, use yp = Aekx .
3. If G(x) = C sin kx or C cos kx, use yp = A cos kx + B sin kx
4. If G(x) is a product of functions, multiply them for yp.
Example: G(x) = x cos 3x yp = (Ax + B) cos 3x + (Cx + D) sin 3x
5. If G(x) is a sum of functions, find separate particular solutions and add them together at the end.
Example: G(x) = xex + cos 2x
Use yp1 = (Ax + B)ex and yp2 = C cos 2x + D sin 2x
Then add y(x) = yc + yp1 + yp2
6. If yp is a solution to the complementary equation (yc), multiply yp by x or x2, so yc and yp are linearly independent.
7. The particular solutions we try to find using yp (the ones with the A, B, C, etc. in them) are called “trial solutions”.
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