Let y = y(x) be the solution curve of the differential equation dy/dx = y/x( 1 + xy²(1 + log ₑx) ),x > 0,y(1) = 3. Then y²(x)/9 is equal to :

Solution

Maths sir · 5 steps

Spot x dy - y dx hiding inside, set t = x/y, and the variables separate cleanly.

1. Spot the grouping. Multiply out and multiply by x: x dy - y dx = x y³(1+ln x) dx. The left side is a known pattern: since d(x/y) = (y dx - x dy)/y², we have x dy - y dx = -y² d(x/y).
2. Separate. Dividing by y² gives -d(x/y) = x y(1+ln x) dx. Put t = x/y, so y = x/t and xy = x²/t. Multiplying across by t separates the variables.

   -t dt = x²(1+ln x) dx

3. Integrate. By parts, ∫ x²ln x dx = (x³ln x)/3 - x³/9, so the right side integrates to 2x³/9 + (x³ln x)/3. Multiplying by -2 gives x²/y² = C - 4x³/9 - (2x³ln x)/3.
4. Use y(1) = 3. At x = 1: 1/9 = C - 4/9, so C = 5/9. Then x²/y² = (5 - 4x³ - 6x³ln x)/9 = (5 - 2x³(2 + 3 ln x))/9, and 3 ln x = ln x³.
5. So. Flipping both sides, y²/9 = x²/(5 - 2x³(2 + ln x³)), which is option (a).

The idea to keep. When you see y/x plus a power of y, multiply out and look for x dy - y dx = -y² d(x/y). The substitution t = x/y then usually separates the variables.

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