Question

The degree of the differential equation \( \int \frac{x}{(x-1)(x-2)^2} dx = a \log \left| \frac{x-1}{x-2} \right| + \frac{b}{x-2} + c \) is:

• A. \( a = -1, b = 2 \)
• B. \( a = -1, b = -2 \)
• C. \( a = 1, b = -2 \)
• D. \( a = 1, b = 2 \)

Hint:

When using partial fractions, remember to solve for unknown coefficients by equating the powers of \( x \) from both sides of the equation.

Answer 1

The Correct Option is C

Step 1: Break the fraction using partial fractions.
We express \( \frac{x}{(x-1)(x-2)^2} \) as \( \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{(x-2)^2} \) and solve for \( A, B, C \).

Step 2: Set up the equation for partial fractions.
Multiply both sides by \( (x-1)(x-2)^2 \) and expand the equation.

Step 3: Compare coefficients.
Compare the coefficients of \( x^2 \), \( x \), and constant terms on both sides of the equation.

Step 4: Solve the system of equations.
Solve for \( A, B, C \) using the system of equations derived from the comparison of coefficients.

Step 5: Find the values of \( A \), \( B \), and \( C \).
We find that \( A = 1 \), \( B = -1 \), and \( C = 2 \).

Step 6: Conclusion.
The degree of the differential equation is given by \( a = 1, b = -2 \), so the correct answer is option (C).