Let y=y(x) be the solution of the differential equation (d y)/(d x)+5/x(x⁵+1) y=(x⁵+1)²/x⁷, x > 0. If y(1)=2, then y(2) is equal to :

Solution

Maths sir · 4 steps

Split 5/x(x⁵+1) into 5/x-5x⁴/(x⁵+1) and the integrating factor falls out neatly.

1. Integrating factor. Write 5/x(x⁵+1) = 5/x - 5x⁴/(x⁵+1). Its integral is 5 ln x - ln(x⁵+1), so the integrating factor is x⁵/(x⁵+1).

   y·x⁵/(x⁵+1) = ∫ (x⁵+1)²/x⁷·x⁵/(x⁵+1) dx = ∫(x³ + 1/x²)dx

2. Integrate. The integrand simplifies to (x⁵+1)/x² = x³ + x⁻². So x⁵/(x⁵+1) y = x⁴/4 - 1/x + C.
3. Use y(1) = 2. At x = 1 the left side is 1/2 × 2 = 1. Then 1 = 1/4 - 1 + C, which gives C = 7/4.
4. So. At x = 2: 32/33 y = 4 - 1/2 + 7/4 = 21/4. So y(2) = 21/4 × 33/32 = 693/128, which is option (a).

The idea to keep. When P(x) looks like k/x(xⁿ+1), split it as k/x - kxⁿ⁻¹/(xⁿ+1). The integrating factor then comes out as a clean ratio of powers.

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