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 $\alpha x+\beta y+3 \log _{e}|2 x+3 y-\gamma|=6$, then $\alpha+2 \beta+3 \gamma$ is equal to ______.

JEE Main · 2024 · 27 Jan 2024 · Shift 1 · Differential Equations

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 lo g_{e} ∣2 x + 3 y - γ ∣ = 6

, then

α + 2 β + 3 γ

is equal to ______.

Answer: 29

Let

t = 2 x + 3 y - 2

, so

4 x + 6 y - 7 = 2 t - 3

and

\frac{d t}{d x} = 2 + 3 \frac{d y}{d x}

. The equation becomes

\frac{d y}{d x} = \frac{- t}{2 t - 3} \Rightarrow \frac{d t}{d x} = 2 - \frac{3 t}{2 t - 3} = \frac{t - 6}{2 t - 3}

Separating and integrating:

\int \frac{2 t - 3}{t - 6} d t = \int d x \Rightarrow 2 t + 9 ln ∣ t - 6∣ = x + c

.

Substituting back,

2 (2 x + 3 y - 2) + 9 ln ∣2 x + 3 y - 8∣ = x + c

. Using

y (0) = 3

gives

c = 14

, and simplifying:

x + 2 y + 3 ln ∣2 x + 3 y - 8∣ = 6

α = 1, β = 2, γ = 8

, and

α + 2 β + 3 γ = 1 + 4 + 24 = 29

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