Let α|x|=|y| e^(x y-β), α, β ∈ N be the solution of the differential equation x d y-y d x+x y(x d y+y d x)=0,y(1)=2. Then α+β is equal to ________

Solution

Maths sir · 4 steps

Divide the whole equation by xy: every piece becomes an exact differential you can integrate directly.

1. Spot the pieces. The bracket x dy + y dx is exactly d(xy). Divide the whole equation by xy so the first two terms also become clean differentials.

   dy/y - dx/x + d(xy) = 0

2. Integrate. This gives ln|y| - ln|x| + xy = C. Put in x = 1, y = 2: C = ln 2 + 2.
3. Remove the logs. ln|y/x| = ln 2 + 2 - xy, so |y| = 2|x| e²⁻ˣʸ. Multiply both sides by eˣʸ⁻² to get 2|x| = |y| eˣʸ⁻².
4. So. Matching with α|x| = |y| e^(xy-β) gives α = 2 and β = 2, so α + β = 4.

The idea to keep. When you see x dy + y dx or x dy - y dx, think d(xy), d(y/x) or ln forms. Divide by the right factor and the equation integrates term by term.

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