Let the solution curve y = y(x) of the differential equation
(4 + x²)dy - 2x(x² + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _____________.

Solution

Maths sir · 4 steps

Divide by 4+x² and the x²+4 on the right cancels. Then it is a clean linear equation in y.

1. Linear form. Divide through by (4+x²) dx. The right side 2x(x²+4)/(4+x²) becomes just 2x, so dy/dx - 6x/(4+x²) y = 2x.
2. Integrating factor. ∫ -6x/(4+x²) dx = -3 ln(4+x²), so the integrating factor is (4+x²)⁻³.

   y/(4+x²)³ = ∫ 2x/(4+x²)³ dx = -1/2(4+x²)² + C

3. Use the origin. Multiply out: y = -1/2(4+x²) + C(4+x²)³. At x=0, y=0 gives -2 + 64C = 0, so C = 1/32.
4. So. At x=2, 4+x² = 8: y(2) = -1/2(8) + 1/32(512) = -4 + 16 = 12.

The common slip: -4. Taking C = 0 because the curve passes through the origin gives -4. The origin condition fixes C = 1/32, not 0.

The idea to keep. Before you hunt for an integrating factor, see whether something on the right side cancels with the coefficient of dy/dx. A neat cancellation usually means the question was designed to be linear.

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