Let f:(0, ∞) → R be a function which is differentiable at all points of its domain and satisfies the condition x² f′(x)=2 x f(x)+3, with f(1)=4. Then 2 f(2) is equal to :

Solution

Maths sir · 4 steps

Divide by x² to get standard linear form, and the integrating factor 1/x² does the rest.

1. Standard form. Divide both sides by x² to get f'(x) - 2/x f(x) = 3/x². This is linear in f, with P = -2/x and Q = 3/x².
2. Integrating factor. ∫ P dx = -2 ln x, so the integrating factor is e⁻²ˡⁿˣ = 1/x². Multiplying by it turns the left side into an exact derivative.

   d/dx(f(x)/x²) = 3/x⁴ ⇒ f(x)/x² = -1/x³ + C

3. Find C. Put x = 1: 4 = -1 + C, so C = 5. Then f(x) = 5x² - 1/x.
4. So. f(2) = 20 - 1/2 = 39/2, so 2f(2) = 39, which is option (d).

The idea to keep. Always bring a linear equation to the form y' + Py = Q before finding the integrating factor. Here P = -2/x gives the factor 1/x², not x².

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