Exams Joint Entrance Examination Mathematics Differential Equations Question 87 Question 87 of 490 Medium NUMERICAL
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 (2x+3y-2)dx+(4x+6y-7)dy=0,y(0)=3 ( 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 \alpha x+\beta y+3{\log }_{e}|2x+3y-\gamma |=6 α x + β y + 3 log e ∣2 x + 3 y − γ ∣ = 6 , then α + 2 β + 3 γ \alpha +2\beta +3\gamma α + 2 β + 3 γ is equal to \_\_\_\_\_\_\_\_\_\_\_\_.
Answer: 29
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2 x + 3 y − 2 = t 4 x + 6 y − 4 = 2 t 2 + 3 d y d x = d t d x 4 x + 6 y − 7 = 2 t − 3 \begin{gathered} 2x+3y-2=t 4x+6y-4=2t \\ 2+3\frac{dy}{dx}=\frac{dt}{dx} 4x+6y-7=2t-3 \end{gathered} 2 x + 3 y − 2 = t 4 x + 6 y − 4 = 2 t 2 + 3 d x d y = d x d t 4 x + 6 y − 7 = 2 t − 3
d y d x = − ( 2 x + 3 y − 2 ) 4 x + 6 y − 7 d t d x = − 3 t + 4 t − 6 2 t − 3 = t − 6 2 t − 3 ∫ 2 t − 3 t − 6 d t = ∫ d x ∫ ( 2 t − 12 t − 6 + 9 t − 6 ) ⋅ 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 α = 1 , β = 2 , γ = 8 α + 2 β + 3 γ = 1 + 4 + 24 = 29 \begin{gathered} \frac{dy}{dx}=\frac{-(2x+3y-2)}{4x+6y-7} \\ \frac{dt}{dx}=\frac{-3t+4t-6}{2t-3}=\frac{t-6}{2t-3} \\ \int \frac{2t-3}{t-6}dt=\int dx \\ \int (\frac{2t-12}{t-6}+\frac{9}{t-6})\cdot dt=x \\ 2t+9\ln (t-6)=x+c \\ 2(2x+3y-2)+9\ln (2x+3y-8)=x+c \\ x=0,y=3 \\ c=14 \\ 4x+6y-4+9\ln (2x+3y-8)=x+14 \\ x+2y+3\ln (2x+3y-8)=6 \\ \alpha =1,\beta =2,\gamma =8 \\ \alpha +2\beta +3\gamma =1+4+24=29 \end{gathered} d x d y = 4 x + 6 y − 7 − ( 2 x + 3 y − 2 ) d x d t = 2 t − 3 − 3 t + 4 t − 6 = 2 t − 3 t − 6 ∫ t − 6 2 t − 3 d t = ∫ d x ∫ ( t − 6 2 t − 12 + t − 6 9 ) ⋅ 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 α = 1 , β = 2 , γ = 8 α + 2 β + 3 γ = 1 + 4 + 24 = 29 Answer the question to unlock