If a curve y = f(x) passes through the point (1, 2) and satisfies x dy/dx + y = bx⁴, then for what value of b, ∫₁² f(x)dx = 62/5?

Solution

Maths sir · 5 steps

The left side is exactly the derivative of xy, so integrate once and the whole curve falls out.

1. Spot the derivative. By the product rule, xdy/dx + y = d/dx(xy). So the equation becomes d/dx(xy) = bx⁴, and one integration gives xy = bx⁵/5 + C.
2. Use the point. Put x = 1, y = 2: 2 = b/5 + C, so C = 2 - b/5. Dividing by x gives f(x) = bx³/5 + C/x.

   ∫₁² f(x) dx = b/20(2⁴ - 1) + C ln 2 = 3b/4 + (2 - b/5)ln 2

3. Match the value. The target 62/5 is rational, but ln 2 is not. So the ln 2 term must vanish: 2 - b/5 = 0 gives b = 10. Check: (3 × 10)/4 = 15/2, and that does not equal 62/5.
4. Recheck the power. Go back to xy = bx⁵/5 + C: dividing by x gives y = bx⁴/5 + C/x, not x³. The integral is then b/25(2⁵ - 1) + C ln 2 = 31b/25 + (2 - b/5)ln 2.
5. So. With b = 10, C = 0 and the integral is (31 × 10)/25 = 62/5, exactly as needed. The answer is (b) 10.

The idea to keep. Whenever you see x y' + y, read it as d/dx(xy) and integrate directly. If a ln term appears in a value that should be rational, its coefficient must be zero.

Answer from the NTA final answer key. Solution by PrepWiser.