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Section 7.5 Method of Undetermined Coefficients

When solving a non-homogeneous linear differential equation, selecting the correct form for the particular solution, yp, is just the beginning. The next step is determining the values of its unknown coefficients, such as A and B. In this section, we’ll explore how to compute these coefficients by substituting the guessed particular solution into the original equation.
Let’s revisit the example we discussed earlier. Consider the differential equation:
(38)y4y+3y=9x.
We guessed that the particular solution would be linear, so we set
yp=Ax+B.
Our goal now is to determine the values of A and B. To do this, we substitute yp=Ax+B into the original differential equation. Here’s how it works:
yp=Ax+Byp=Ayp=0
Substituting yp, yp, and yp into (38) results in:
04A+3(Ax+B)= 9x4A+3Ax+3B= 9x
By matching the coefficients of like terms on opposite sides of the equation,
we solve:
3A= 9A=34A+3B= 0B=4
Thus, the particular solution is:
yp=3x+4.
Regardless of yp’s form, the process of finding the coefficients remains consistent:
  • Substitute the chosen yp into the differential equation.
  • Collect and match coefficients of like terms on both sides of the equation.
  • Solve the resulting system to find A,B,C,.
This is why the method is called the "method of undetermined coefficients." and the complete process is summarized as follows.

Method 5. Undetermined Coefficients.

The linear nonhomogeneous equation with constant coefficients,
(39)an y(n)+an1 y(n1)++a2 y+a1 y+a0 y=f(x),
has a general solution given by
(40)y=yh+yp,
where yh  is the solution to the homogeneous equation
(41)an y(n)+an1 y(n1)++a2 y+a1 y+a0 y=0,
and yp is the particular solution found through the following steps...
Select Initial yp
Select the initial form of yp that generalizes f(x).
Adjust yp
If yp shares terms with yh, repeatedly multiply the terms by x until there are no terms in common.
Find the Coefficients of yp
Plug yp into (39) and match terms on each side to solve for the unknown coefficients of yp.
Now for a few examples to illustrate the complete application of the nndetermined coefficients method.

Example 22. Find the general solution for each equation.

Solution 1. y3y+y=2x2+3x
Following the method of Undetermined Coefficients, we first find yh as
yh=c1ex+c2xex.
yp=Ax2+Bx+C
and adjust yp, if necessary. A quick check shows that yp has no terms in common with yh.
Finally, we find the coefficients of yp. To do this, let’s compute yp, yp, plug these into the equation, and collect the x2, x, and constant terms as follows:
yp=2Ax+Byp=2A
2A3(2Ax+B)+(Ax2+Bx+C)=2x2+3x2A6Ax3B+Ax2+Bx+C=2x2+3xA1x2+(6A+B)2x+(2A3B+C)3=21x2+32x+03.
Matching the underlined coefficients leads to the following system of equations and solution for the coefficients A, B, and C:
 1. 2. 3.
A= 26A+B= 32A3B+C= 0
 A=2 B=9 C=13
The general solution is then
y=yh+yp=c1ex+c2xex+2x2+9x+13.
Solution 2. y+3y28y=7t+e4t1
Following the method of Undetermined Coefficients, we find yh as yh=c1e4t+c2e7t and then find yp through the steps that follow.
Select Initial yp. The initial form of yp is
yp=At+B+Ce4t,
Adjust yp. Notice that yp has an e4t term, which overlaps with yh. Therefore, we need to adjust yp by multiplying the e4t term by t,
yp=At+B+Cte4t.
Now that yp and yh are independent, we can proceed to find the coefficients A, B, and C.
Find the Coefficients of yp. Before plugging in yp, let’s find yp and yp:
yp= At+B+Cte4typ= A+Ce4t+4Cte4typ= 8Ce4t+16Cte4t.
Now, substituting these into the equation and collecting like terms, to get
8Ce4t+16Cte4t +3(A+Ce4t+4Cte4t) 28(At+B+Cte4t)=7t+e4t1(11C)1et+(28A)2t+(3A28B)3=(1)1e4t+(7)2t+(1)3,
which leads to the following equations for A, B, and C:
 1. 2. 3.
11C= 128A= 73A28B= 1
 C=1/11 A=1/4 B=1/112
Finally, the general solution is
y=yh+yp=c1e4t+c2e7t14t+1112+111e4t.

Reading Questions Check-Point Questions

1. After substituting yp=Ax+B into the equation y4y+3y=9x, you find the system of equations...

    After substituting yp=Ax+B into the equation y4y+3y=9x, you find the system of equations:
    3A=9
    and
    4A+3B=0.
    What are the values of A and B?
  • A=3,B=4
  • Correct! Solving the system yields A=3 and B=4.
  • A=1,B=2
  • Incorrect. Double-check your algebra when solving the system of equations.
  • A=3,B=0
  • Incorrect. While A is correct, you need to solve for B as well.
  • A=2,B=4
  • Incorrect. A=3, not 2.

2. Which of the following is NOT a step in finding the coefficients of the particular solution?

    Which of the following is NOT a step in finding the coefficients of the particular solution?
  • Plugging yp into the differential equation.
  • Incorrect. This is the first step in determining the coefficients.
  • Applying the initial conditions.
  • Correct! The initial conditions are used to solve for the unknown constants in the general solution, not the particular solution.
  • Collecting and matching like terms.
  • Incorrect. This is a crucial step for finding the values of A and B.
  • Solving the resulting system of equations.
  • Incorrect. This is the final step in finding the coefficients.

3. Consider the equation...

    Consider the equation
    (42)y4y+3y=9x.
    If the particular solution is assumed to be
    yp=Ax+B,
    which of the following represents the correct coefficient-matching equations when substituting into the differential equation?
  • 3A=9,4A+3B=9
  • Incorrect. Remember to match coefficients of like terms separately (i.e., coefficients of x and constant terms).
  • 3A=9,4A+3B=0
  • Correct! Matching coefficients of x gives 3A=9, and matching constant terms gives 4A+3B=0.
  • A=9,4A+B=0
  • Incorrect. Re-check the coefficients and ensure you match the terms correctly.
  • 3A=3,4B+A=9
  • Incorrect. Verify which terms correspond to coefficients of x and the constant terms.

4. Why do we adjust the form of yp by multiplying terms by x when there is overlap with yh?

    Why do we adjust the form of yp by multiplying terms by x when there is overlap with yh?
  • To make the computation of derivatives easier.
  • Incorrect. The adjustment is not made for the ease of differentiation but to ensure the independence of terms.
  • To ensure that yp and yh remain independent.
  • Correct! Multiplying by x ensures that the particular solution does not overlap with the homogeneous solution.
  • To match the degree of the polynomial in f(x).
  • Incorrect. While the degree of yp is related to f(x), the adjustment addresses overlap with yh.
  • To simplify the resulting system of equations.
  • Incorrect. The adjustment is made to maintain independence between yp and yh, not for simplification purposes.

5. Suppose
y4y+3y=6ex.
Which of the following is the correct initial guess for yp?

    Suppose
    y4y+3y=6ex.
    Which of the following is the correct selection for yp?
  • yp=Aex
  • Incorrect. Remember to consider whether ex overlaps with any terms in yh.
  • yp=Axex
  • Correct! Since ex is a term in yh, we need to multiply the particular solution by x to ensure independence.
  • yp=Ax+B
  • Incorrect. This form is more suited for a polynomial right-hand side like f(x)=6x.
  • yp=A
  • Incorrect. A constant form of yp would only be appropriate if f(x) were a constant.

6. What are the main steps for finding yp using the method of undetermined coefficients?

    What are the main steps for finding yp using the method of undetermined coefficients?
  • Select the particular solution, integrate it, and solve for coefficients.
  • Incorrect. Integration is not a part of the method of undetermined coefficients.
  • Select the particular solution, adjust it if necessary, and find its coefficients.
  • Correct! The process involves making an educated guess, substituting it, and solving for unknown coefficients.
  • Select the homogeneous solution, adjust for yp, and differentiate.
  • Incorrect. The focus is on selecting and solving for yp, not guessing the homogeneous solution.
  • Choose yp, integrate the result, and verify the solution.
  • Incorrect. Integration is not a step in this method; it’s about solving for coefficients of yp.
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