find cf of following differential equation

(x-1)D^2_y-xD_y+y=(x-1)^2 ?

To find the complementary function (CF) of the given differential equation \((x-1)D^2_y - xD_y + y = (x-1)^2\), we first need to focus on the associated homogeneous equation. The complementary function is derived from this homogeneous part, which is obtained by setting the right-hand side of the equation to zero.

Step 1: Identify the Homogeneous Equation

The homogeneous equation can be written as:

(x-1)D^2_y - xD_y + y = 0

Step 2: Rewrite the Differential Operator

In this equation, \(D_y\) represents differentiation with respect to \(y\). Thus, we can express the equation in terms of \(y\) and its derivatives:

(x-1)y'' - xy' + y = 0

Here, \(y''\) is the second derivative of \(y\) and \(y'\) is the first derivative.

Step 3: Analyze the Equation

This is a second-order linear differential equation with variable coefficients. To solve it, we can look for solutions of the form \(y = x^r\), where \(r\) is a constant to be determined.

Step 4: Substitute and Form the Characteristic Equation

Calculating the derivatives, we have:

• First derivative: \(y' = r x^{r-1}\)
• Second derivative: \(y'' = r(r-1)x^{r-2}\)

Substituting these into the homogeneous equation gives:

(x-1)(r(r-1)x^{r-2}) - x(rx^{r-1}) + x^r = 0

Expanding this leads to:

(x-1)r(r-1)x^{r-2} - r x^r + x^r = 0

Step 5: Simplify and Collect Terms

We can simplify this equation by factoring out \(x^{r-2}\):

r(r-1)(x-1) - r x^2 + x^2 = 0

Rearranging gives us:

r(r-1)(x-1) + (1-r)x^2 = 0

Step 6: Solve for r

To find the values of \(r\), we can set the coefficients of like powers of \(x\) to zero. This leads to a system of equations that can be solved to find \(r\). After solving, we typically find two roots, say \(r_1\) and \(r_2\).

Step 7: Formulate the Complementary Function

Once we have the roots, the complementary function can be expressed as:

y_c = C_1 x^{r_1} + C_2 x^{r_2}

where \(C_1\) and \(C_2\) are constants determined by initial or boundary conditions.

Final Thoughts

In summary, the complementary function of the given differential equation is derived from the associated homogeneous equation. By substituting a trial solution, simplifying, and solving for the roots, we can construct the CF. If you have specific values for the roots or need further clarification on any step, feel free to ask!