Solving Non-Homogeneous Differential Equations with the Method of Undetermined Coefficients

Non-homogeneous differential equations often arise in various engineering and physics problems. This article focuses on a specific example to demonstrate how to solve the differential equation y#x2032;#x2212;2y#x2033;#x2032;#x2212;3yexcosx using the method of undetermined coefficients. We will walk through the process of finding the general solution by breaking it down into three steps: finding the complementary solution, determining a particular solution, and combining these solutions.

Step 1: Complementary Solution

First, we start by solving the homogeneous part of the differential equation:

y#x2032;#x2212;2y#x2033;#x2032;#x2212;3y0

The characteristic equation for this homogeneous equation is:

r2#x2212;2r#x2212;30

Factoring this equation, we get:

(r#x2212;3)(r 1)0

The roots of the characteristic equation are (r 3) and (r -1). Therefore, the complementary solution is:

y_cC1e#x2212;3x C2ex

where (C_1) and (C_2) are arbitrary constants.

Step 2: Particular Solution

Now, to find a particular solution for the non-homogeneous part (e^x cos x), we will use the method of undetermined coefficients. The suggested trial solution is:

y_pex(Acosx Bsinx)

We differentiate this trial solution to find the first and second derivatives:

First derivative:

y_pex(Acosx Bsinx)(1cosx Bsinx)

Second derivative:

y_p#x2033;ex(2Acosx 2Bsinx)

Substitute these into the original differential equation:

ex(2Acosx 2Bsinx)#x2212;2ex(Acosx Bsinx)#x2212;3ex(Acosx Bsinx)excosx

Simplifying and combining like terms, we get:

ex(2A#x2212;2A#x2212;3Acosx 2B#x2212;2B#x2212;3Bsinx)excosx

Setting the coefficients equal, we have:

For (cos x): ((2A - 2A - 3A) 1) which simplifies to (3A 1), thus (A frac{1}{3})

For (sin x): ((2B - 2B - 3B) 0), thus (B -frac{1}{3})

So the particular solution is:

y_pex(13cosx#x2212;13sinx)

Step 3: General Solution

Finally, the general solution is the sum of the complementary and particular solutions:

yy_c y_pC_11e#x2212;3x C_22ex 13excosx#x2212;13exsinx

Therefore, the general solution to the differential equation is:

boxed{y C_1 e^{-3x} C_2 e^{x} frac{1}{3} e^x cosx frac{1}{3} e^x sinx}

The method of undetermined coefficients, while powerful, can be a bit complex. However, it greatly simplifies the process of finding particular solutions to differential equations with non-homogeneous parts, especially when the form is relatively simple as in this example with (e^x cos x).