Let y = y(x) be the solution of the differential equation x(1 - x²)dy/dx + (3x²y - y - 4x³) = 0, x > 1, with y(2) = - 2. Then y(3) is equal to :

Solution

Maths sir · 5 steps

Write it in linear form: P is -f'/f with f = x - x³, so the integrating factor is 1/f.

1. Standard form. Divide by x(1-x²) and move 4x³ to the right. This gives dy/dx + (3x²-1)/(x-x³) y = 4x²/(1-x²).
2. Integrating factor. The derivative of x - x³ is 1 - 3x², so P = -(x-x³)'/(x-x³). That makes ∫ P dx = -ln(x³ - x) for x > 1, and the integrating factor is 1/(x³-x).

   d/dx(y/(x³-x)) = 4x²/((1-x²) x(x²-1)) = -4x/(x²-1)²

3. Integrate. Put t = x² - 1. Then ∫ -4x/(x²-1)² dx = 2/(x²-1) + C. Multiply through by x³ - x = x(x²-1) to get y = 2x + C(x³ - x).
4. Use the condition. At x = 2, y = -2: 4 + 6C = -2 gives C = -1. So y = 2x - x³ + x = 3x - x³.
5. So. y(3) = 9 - 27 = -18, which is option (a).

The idea to keep. In a linear DE, always check whether P is a derivative over its function. If P = ± f'/f, the integrating factor is f or 1/f straight away. Watch the sign.

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