17. Method of Undetermined Coefficients (Superposition Approach)

Dated: 30-11-2024

Recall

The non homogeneous linear differential equation of order \(n\) is an equation of the form

\[a_n\frac{d^ny}{dx^n}+a_{n-1}\frac{d^{n-1}y}{dx^{n-1}}+\ldots+a_1\frac{dy}{dx}+a_0y=g(x)\]

To obtain the general solution, we obtain
- The complementary function \(y_c\) which is general solution to the associated homogeneous equation.
- Any particular solution \(y_p\) to the non homogeneous differential equation.
Then

\[\text{General solution} = \text{Complementary function} + \text{Particular Integral}\]

Methods for Finding Integral for Non Homogeneous Equation

The method of undetermined coefficients-superposition approach
The method of undetermined coefficients-annihilator operator approach.
The method of variation of parameters.

The Method of Undetermined Coefficients-superposition Approach

The method developed here is limited to non homogeneous linear differential equations with following factors.

Constant coefficients
\(g(x)\) has a specific form

Form of \(g(x)\)

Following forms are valid for this method

A constant function \(k\)
A polynomial function
An exponential function \(e^x\)
The trignometric functions \(\sin(\beta x), \cos(\beta x)\)
Finite sum or products of these functions.

Method

Determine the form of \(g(x)\).
Assume the general form of \(y_p\) according to form of \(g(x)\).
Substitute in the given non-homogeneous differential equation.
Simplify and equate coefficients of like terms from both sides.
Solve the resulting equations to find the unknown coefficients.
Substitute the calculated values of coefficients in assumed \(y_p\).

Reasons behind Restrictions on \(g(x)\)

The derivatives of sums or products of polynomials, exponentials are again sums and products of similar kind of functions.
The expression \(ay_p^{\prime\prime} + by_p^\prime + cy_p\) has to be identically equal to the input function \(g(x)\).

Caution

No function in assumed \(y_p\) should contain a solution to the associated homogeneous differential equation.
This means that \(y_p\) should not contain terms which duplicate terms in \(y_c\).

The input function \(g(x)\)	The assumed particular solution \(y_p\)
Any constant e.g. 1	\(A\)
\(5x+7\)	\(Ax+B\)
\(3x^2-2\)	\(Ax^2+Bx+c\)
\(x^3-x+1\)	\(Ax^3+Bx^2+Cx+D\)
\(\sin 4x\)	\(A\cos 4x+B\sin 4x\)
\(\cos 4x\)	\(A\cos 4x+B\sin 4x\)
\(e^{5x}\)	\(Ae^{5x}\)
\((9x-2)e^{5x}\)	\((Ax+B)e^{5x}\)
\(x^2e^{5x}\)	\((Ax^2+Bx+C)e^{5x}\)
\(e^{3x}\sin 4x\)	\(Ae^{3x}\cos 4x+Be^{3x}\sin 4x\)
\(5x^2\sin 4x\) 1	\((A_1x^2+B_1x+C_1)\cos 4x+(A_2x^2+B_2x+C_2)\sin 4x\)
\(xe^{3x}\cos 4x\)	\((Ax+B)e^{3x}\cos 4x+(Cx+D)e^{3x}\sin 4x\)

If \(g(x)\) Equals a Sum

If

\[g(x) = g_1(x) + g_2(x) + \ldots + g_m(x)\]

Then

\[y_p = y_{p_1} + y_{p_2} + \ldots + y_{p_m}\]

The form of \(y_p\) is the linear combination of all the linearly independent functions generated by the repeated differentiation of the input function \(g(x)\).

Example

\[y^{\prime\prime} + 4y^\prime - 2y = 2x^2 - 3x + 6\]

Complementary Function

To find \(y_c\), we solve

\[y^{\prime\prime} + 4y^\prime - 2y = 0\]

\[y = e^{mx}, \quad y^\prime = me^{mx}, \quad y^{\prime\prime} = m^2e^{mx}\]

Then the associated homogeneous equation gives

\[(m^2 + 4m - 2)e^{mx} = 0\]

Therefore, the auxiliary equation is

\[m^2 + 4m - 2 = 0 \quad \text{as} \quad e^{mx} \neq 0, \ \forall \ x\]

Using the quadratic formula, roots of the auxiliary equation are

\[m = -2 \pm \sqrt{6}\]

Thus we have real and distinct roots of the auxiliary equation

\[m_1 = -2 - \sqrt{6} \quad \text{and} \quad m_2 = -2 + \sqrt{6}\]

Hence the complementary function is

\[y_c = c_1 e^{(-2 - \sqrt{6})x} + c_2 e^{(-2 + \sqrt{6})x}\]

Next, we find a particular solution of the non-homogeneous differential equation.

Particular Integral

Since the input function

\[ g(x) = 2x^2 - 3x + 6 \]

is a quadratic polynomial. Therefore, we assume that

\[ y_p = Ax^2 + Bx + C \]

Then

\[ y_p' = 2Ax + B \quad \text{and} \quad y_p'' = 2A \]

Therefore

\[ y_p'' + 4y_p' - 2y_p = 2A + 8Ax + 4B - 2Ax^2 - 2Bx - 2C \]

Substituting in the given equation, we have

\[ 2A + 8Ax + 4B - 2Ax^2 - 2Bx - 2C = 2x^2 - 3x + 6 \]

or

\[ -2Ax^2 + (8A - 2B)x + (2A + 4B - 2C) = 2x^2 - 3x + 6 \]

Equating the coefficients of the like powers of x, we have

\[ -2A=2, \quad 8A-2B=-3, \quad 2A+4B-2C=6 \]

Solving this system of equations leads to the values

\[ A=-1, \quad B=-\frac{5}{2}, \quad C=-9. \]

Thus a particular solution of the given equation is

\[ y_p = -x^2 - \frac{5}{2}x - 9. \]

Hence, the general solution of the given non-homogeneous differential equation is given by

\[ y = y_c + y_p \]

or

\[ y = -x^2 - \frac{5}{2}x - 9 + c_1 e^{(-2+\sqrt{6})x} + c_2 e^{(-2-\sqrt{6})x} \]

Duplication between \(y_p\) and \(y_c\)

Assuming

\[y_p = y_{p_1} + y_{p_2} + \ldots + y_{p_n}\]

and there are terms in \(y_{p_i}\) which duplicate terms in \(y_c\) then \(y_{p_i}\) must be multiplied with \(x^n\) where \(n\) is the least positive integer that eliminates the duplication.