If the solution of the differential equation
(2 x+3 y-2) d x+(4 x+6 y-7) d y=0, y(0)=3, is
α x+β y+3 log ₑ|2 x+3 y-γ|=6, then α+2 β+3 γ is equal to ____________.

Solution

Maths sir · 5 steps

The same 2x+3y sits in both brackets, so call it t and the equation separates.

1. Substitute. Put t = 2x + 3y, so dt/dx = 2 + 3dy/dx. The equation gives dy/dx = -(t-2)/(2t-7).
2. Separate. Then dt/dx = 2 - 3(t-2)/(2t-7) = (t-8)/(2t-7). Split the fraction: (2t-7)/(t-8) = 2 + 9/(t-8).

   ∫(2 + 9/(t-8))dt = ∫ dx ⇒ 2t + 9 ln|t-8| = x + C

3. Substitute back. Put t = 2x + 3y: 4x + 6y - x + 9 ln|2x+3y-8| = C. Divide by 3 to match the given form: x + 2y + 3 ln|2x+3y-8| = C'.
4. Use the condition. At x = 0, y = 3: 2x + 3y - 8 = 1 and ln 1 = 0, so C' = 6. This matches the given right side, so α = 1, β = 2, γ = 8.
5. So. α + 2β + 3γ = 1 + 4 + 24 = 29.

The common slip: 39. Not dividing by 3 leaves 3x + 6y + 9 ln|2x+3y-8| = 18. Reading α = 3, β = 6 from that gives 39. Scale it so the log coefficient is 3, as in the given form.

The idea to keep. When dx and dy carry the same linear combination like ax+by, substitute t = ax+by. The equation becomes variable separable in t and x.

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