# Method of Undetermined Coefficients

The general solution to the linear differential equation L(y)=phi(x) is given as y=y_h+y_p, where y_p denotes one solution to the differential equation and y_h is the general solution to the associated homogeneous equation L(y) = 0. For the method to obtain y_h when the differential equation has constant coefficients, see the method of solutions note.
Here, one method will be given for obtaining the particular solution y_p once y_h is known.

The method of undetermined coefficients is applicable only if phi(x) and all of its derivatives can be written in terms of the same finite set of linearly independent functions, which we denote by {y_1(x),\ y_2(x),\ ...,\ y_n(x)}.
The method is initiated by assuming a particular solution of the form y_p=A_1 y_1(x)+A_2 y_2(x)+...+A_n y_n(x), where A_1,\ A_2, ...,A_n denote arbitrary multiplicative constants. These arbitrary constants are then evaluated by substituting the proposed solution into the given differential equation and equating the coefficients of the like terms.

Case 1. phi(x)=p_n(x), a polynomial of n-th degree. Assume the solution of the form y_p=A_n x^n+A_(n-1) x^(n-1)+...+Ax+A_0, where A_j (j=0, 1, ..., n) are the constants to be determined.

Example 1. Find the particular solution of y''+3y'+2y=x^2+2.

Assume that the particular solution is of the form y_p=Ax^2+Bx+C.

Then, y_p'=2Ax+B, and y_p''=2A.

Now, plug these values into the differential equation:

(2A)+3(2Ax+B)+2(Ax^2+Bx+C)=2Ax^2+(6A+2B)x+(2A+3B+C).

Equating the like terms with x^2+2 gives:

{(2A=1),(6A+2B=0),(2A+3B+C=2):},

Which has the solution A=1/2,B=-3/2,C=11/4.

So, the particular solution is y_p=1/2 x^2-3/2 x+11/4.

Case 2. phi(x)=ke^(alpha x). Assume the solution of the form y_p=Ae^(alpha x), where A is the constant to be determined.

Example 2. Find the particular solution of y'''-4y'+2y=3e^(7x).

Assume that the particular solution is of the form y_p=Ae^(7x).

Then, y_p'=7Ae^(7x), y_p''=49Ae^(7x), and y_p'''=343Ae^(7x).

Plugging these values into the equation gives 343Ae^(7x)-4*7Ae^(7x)+2Ae^(7x)=317Ae^(7x).

Equating the coefficients with 3e^(7x) gives 317A=3, or A=3/317.

So, the particular solution is y_p=3/317 e^(7x).

Case 3. phi(x)=k_1cos(beta x)+k_2 sin(beta x). Assume the solution in the form of y_p=Acos(beta x)+Bcos(beta x) (even when either k_1 or k_2 equals zero).

Example 3. Solve y''+y=cos(2x).

We are asked to solve the equation, so we should first detemine the solution of the homogeneous equation y''+y=0.

The characteristic equation is r^2+1=0 that has the roots r_1=i,\ r_2=-i; so, the solution of the homogeneous equation (see how to solve a homogeneous equation) is y_h=c_1 cos(x)+c_2 sin(x).

Now, assume that the particular solution is of the form y_p=Acos(2x)+Bsin(2x); then, y_p'=-2Asin(2x)+2Bcos(2x), and y_p''=-4Acos(2x)-4Bsin(2x).

Plugging these values into the equation gives: -4Acos(2x)-4Bsin(2x)+Acos(2x)+Bsin(2x)=-3Acos(2x)-3Bsin(2x).

Equating the like terms with cos(2x), we obtain that

{(-3A=1),(-3B=0):}

So, A=-1/3, B=0.

Thus, the particular solution is y_p=-1/3cos(2x).

Finally, the general solution is y=y_h+y_p=c_1 cos(x)+c_2 sin(2x)-1/3 cos(2x).

Generalizations. If phi(x) is the product of functions from the above cases, assume the particular solution to be the product of the assumed solutions of the corresponding cases. For example, if phi(x)=e^(alpha x)p_n(x)sin(beta x) or phi(x)=e^(alpha x)p_n(x)cos(beta x), assume the particular solution in the form

y_p=e^(alpha x)(A_n x^n+A_(n-1)x^(n-1)+...+A_1x+A_0)cos(beta x)+

+e^(alpha x)(B_n x^n+B_(n-1)x^(n-1)+...+B_1x+B_0)cos(beta x), where A_j and B_j (j=0, 1, ..., n) are constants that still need to be determined.

If phi(x) is the sum (or difference) of the terms already considered, then we take y_p to be the sum (or difference) of the corresponding assumed solutions and algebraically combine the arbitrary constants where possible.

Example 4. Find the particular solution of y''+y=e^(-x)cos(2x)+x^2.

Here, phi(x) is the sum of two manageable functions: phi(x)=(e^(-x)cos(2x))+(x^2).

So, assume the particular solution in the form of y_p=(e^(-x)(Acos(2x)+Bsin(2x)))+(Cx^2+Dx+E).

Then, y_p'=-Ae^(-x)cos(2x)-2Ae^(-x)sin(2x)-Be^(-x)sin(2x)+2Be^(-x)cos(2x)+2Cx+D, and

y_p''=Ae^(-x)cos(2x)+2Ae^(-x)sin(2x)+2Ae^(-x)sin(2x)-4Ae^(-x)cos(2x)+Be^(-x)sin(2x)-2Be^(-x)cos(2x)-2Be^(-x)cos(2x)-4Be^(-x)sin(2x)+2C.

Plugging these values into the equation gives:

Ae^(-x)cos(2x)+2Ae^(-x)sin(2x)+2Ae^(-x)sin(2x)-4Ae^(-x)cos(2x)+Be^(-x)sin(2x)-2Be^(-x)cos(2x)-2Be^(-x)cos(2x)-4Be^(-x)sin(2x)+2C+e^(-x)(Acos(2x)+Bsin(2x))+Cx^2+Dx+E=(A-4A-2B-2B+A)e^(-x)cos(2x)+(2A+2A+B-4B+B)e^(-x)sin(2x)+Cx^2+Dx+(2C+E)=(-2A-4B)e^(-x)cos(2x)+(4A-2B)e^(-x)sin(2x)+Cx^2+Dx+(2C+E)

Equating the like terms with e^(-x)cos(2x)+x^2, we obtain that

{(-2A-4B=1),(4A-2B=0),(C=1),(D=0),(2C+E=0):}

This system has the following solution: A=-1/10, B=-1/5, C=1, D=0,E=-2; so, the particular solution is

y_p=-1/10 e^(-x)cos(2x)-1/5 e^(-x)sin(2x)+x^2-2.

If any term of the assumed solution, disregarding the multiplicative constants, is also a term of y_h (the homogeneous solution), the assumed solution has to be modified by multiplying it by x^m, where m is the smallest positive integer such that the product of x_m with the assumed solution has no terms in common with y_h.

Example 5. Find the particular solution of y''+y=cos(x).

As already known from example 3, the homogeneous solution is y_h=c_1cos(x)+c_2sin(x).

We can't take the particular solution to be Acos(x)+Bsin(x) because its terms are parts of the homogeneous solution; so, let

y_p=Axcos(x)+Bxsin(x).

Then, y_p'=Acos(x)-Axsin(x)+Bsin(x)+Bxcos(x), and y_p''=-Asin(x)-Asin(x)-Axcos(x)+Bcos(x)+Bcos(x)-Bxsin(x).

Plugging these values into the equation gives:

-Asin(x)-Asin(x)-Axcos(x)+Bcos(x)+Bcos(x)-Bxsin(x)+Axcos(x)+Bxsin(x)=-2Asin(x)+2Bcos(x).

Equating the like terms with cos(x), we obtain that

{(-2A=0),(2B=1):}

The solution is A=0, B=1/2.

So, the particular solution is y_p=1/2 xsin(x).

Now, let's take a look at another example.

Example 6. Solve y''-2y'+y=e^(x).

First, find the solution of the corresponding homogeneous equation y''-2y'+y=0.

The characteristic equation is r^2-2r+1=0 or (r-1)^2=0 that has the roots r_1=1, r_2=1.

There is one root of the multiplicity 2; so, the solution is y_h=c_1e^x+c_2 xe^x.

Now, what will be the particular solution? It isn't e^x because e^x is already in the solution, it is not xe^x either. The lowest number m for which x^me^x is not in the solution is m=2: x^2 e^x.

So, let y_p=Ax^2e^x.

Then, y_p'=2Axe^x+Ax^2e^x=Ae^x (x^2+2x), and y_p''=2Ae^x+2Axe^x+2Axe^x+Ax^2e^x=Ae^x (x^2+4x+2).

Plugging the obtained values into equation, we have

Ae^x(x^2+4x+2)-2Ae^x(x^2+2x)+Ax^2 e^x=e^x

2Ae^x=e^x

Equating the like terms with e^x gives 2A=1, or A=1/2.

So, the particular soltuion is y_p=1/2 x^2e^x.

Finally, the general solution is y=y_h+y_p=c_1e^x+c_2xe^x+1/2 x^2 e^x.

As can be seen, the method of undetermined coefficients requires many calculations and solving a system of linear equations. So, it is time-consuming.

In general, if phi(x) is not one of the types of functions considered above or if the differential equation does not have constant coefficients, the method of undetermined coefficients is unapplicable. For example, the particular solution to the equation y''+y=(cos(x))/sin(x) cannot be found with the method of undetermined coefficients.