Find the complementary function of (D^4 + 2D^3 - 3D^2 - 4D + 4)y = e^(-2x) + e^x

To find the complementary function of the differential equation

\[
(D^4 + 2D^3 - 3D^2 - 4D + 4)y = e^{-2x} + e^x,
\]

we begin by solving for the homogeneous equation (the complementary function). The homogeneous equation corresponds to the left-hand side of the equation being equal to 0, i.e.,

\[
(D^4 + 2D^3 - 3D^2 - 4D + 4)y = 0.
\]

### Step 1: Find the characteristic equation
The operator \( D \) represents differentiation with respect to \( x \), so we can rewrite the homogeneous equation in terms of its characteristic equation. We replace each \( D^n \) with \( r^n \) to get:

\[
r^4 + 2r^3 - 3r^2 - 4r + 4 = 0.
\]

### Step 2: Solve the characteristic equation
We need to find the roots of the characteristic equation \( r^4 + 2r^3 - 3r^2 - 4r + 4 = 0 \).

We can try using the Rational Root Theorem to test for possible rational roots. The possible rational roots are the factors of the constant term (4) divided by the factors of the leading coefficient (1). This gives us the possible roots: \( \pm 1, \pm 2, \pm 4 \).

Let's test \( r = 1 \):

\[
1^4 + 2(1^3) - 3(1^2) - 4(1) + 4 = 1 + 2 - 3 - 4 + 4 = 0.
\]

So, \( r = 1 \) is a root.

### Step 3: Factor the characteristic polynomial
Since \( r = 1 \) is a root, we can factor the polynomial \( r^4 + 2r^3 - 3r^2 - 4r + 4 \) as:

\[
(r - 1)(r^3 + 3r^2 - r - 4) = 0.
\]

Now, we need to factor \( r^3 + 3r^2 - r - 4 \). Let's check if \( r = -1 \) is a root:

\[
(-1)^3 + 3(-1)^2 - (-1) - 4 = -1 + 3 + 1 - 4 = -1.
\]

Since \( r = -1 \) is not a root, we can try \( r = 2 \):

\[
2^3 + 3(2^2) - 2 - 4 = 8 + 12 - 2 - 4 = 14.
\]

This is not a root either. However, finding the factors may require some iterative work, checking other values and evaluating the resulting polynomial.



Learn how to find the complementary function of the differential equation (D^4 + 2D^3 - 3D^2 - 4D + 4)y = e^(-2x) + e^x. This step-by-step solution covers how to solve the characteristic equation, apply the Rational Root Theorem, and understand the process involved. Suitable for advanced high school students (Grades 11-12).

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