QJEEM24Mafd31: JEE Main 2024 Mathematics - Class 12 JEEM24Mafd31 Solution: If the solution of the differential equation , is , then is…

$\begin{array}{ll}2 x+3 y-2=t & 4 x+6 y-4=2 t \\2+3 \frac{d y}{d x}=\frac{d t}{d x} & 4 x+6 y-7=2 t-3\end{array}$

$\begin{aligned}& \frac{d y}{d x}=\frac{-(2 x+3 y-2)}{4 x+6 y-7} \\& \frac{d t}{d x}=\frac{-3 t+4 t-6}{2 t-3}=\frac{t-6}{2 t-3} \\& \int \frac{2 t-3}{t-6} d t=\int d x \\& \int\left(\frac{2 t-12}{t-6}+\frac{9}{t-6}\right) \cdot d t=x \\& 2 t+9 \ln (t-6)=x+c \\& 2(2 x+3 y-2)+9 \ln (2 x+3 y-8)=x+c \\& x=0, y=3 \\& c=14 \\& 4 x+6 y-4+9 \ln (2 x+3 y-8)=x+14 \\& x+2 y+3 \ln (2 x+3 y-8)=6 \\& \alpha=1, \beta=2, \gamma=8 \\& \alpha+2 \beta+3 \gamma=1+4+24=29\end{aligned}$