If the solution curve y = y(x) of the differential equation y²dx + (x² - xy + y²)dy = 0, which passes through the point (1, 1) and intersects the line y = √3 x at the point (α ,√3 α ), then value of log ₑ(√3 α ) is equal to :

Solution

Maths sir · 4 steps

Solve for dx/dy instead of dy/dx, put x = vy, and the equation separates at once.

1. Choose x = vy. Write the equation as dx/dy = -(x² - xy + y²)/y², which is homogeneous. Put x = vy, so dx/dy = v + ydv/dy. This gives v + ydv/dy = -(v² - v + 1), which simplifies to ydv/dy = -(1 + v²).
2. Separate and integrate. Separate the variables: dv/(1 + v²) = -dy/y. Integrating both sides gives tan⁻¹ v = -ln y + C.
3. Fix the constant. At (1, 1), v = x/y = 1 and ln 1 = 0. So C = tan⁻¹ 1 = π/4, and the solution curve is:

   tan⁻¹(x/y) + ln y = π/4

4. So. At (α, √3α), x/y = 1/√3, so tan⁻¹ 1/√3 = π/6. Then ln(√3α) = π/4 - π/6 = π/12, which is option (c).

The idea to keep. When the dy term carries the messy quadratic, treat x as the function: put x = vy and solve for dx/dy. The working is much shorter that way.

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