Let y=y(x) be the solution of the differential equation (x y-5 x² √(1+x²)) d x+(1+x²) d y=0, y(0)=0. Then y(√3) is equal to

Solution

Maths sir · 4 steps

This is a linear equation in y: put it in standard form and the integrating factor is √(1+x²).

1. Standard form. Divide by (1+x²) dx to get dy/dx + x/(1+x²) y = 5x²/√(1+x²). This is linear in y with P = x/(1+x²).
2. Integrating factor. ∫ P dx = 1/2 ln(1+x²), so the integrating factor is e^(1/2 ln(1+x²)) = √(1+x²).

   d/dx(y√(1+x²)) = 5x² ⇒ y√(1+x²) = 5x³/3 + C

3. Use y(0) = 0. Putting x = 0, y = 0 gives 0 = 0 + C, so C = 0 and y = 5x³/3√(1+x²).
4. So. At x = √3: x³ = 3√3 and √(1+3) = 2, so y = (5 × 3√3)/(3 × 2) = 5√3/2. The answer is (a).

The idea to keep. When dy has a coefficient and the dx part has y only to the first power, think linear equation. For P = x/(1+x²), the integrating factor is always √(1+x²).

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