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 ________

Correct answer: 0 aligned & a|x|=|y| e^ y x-β , a, b ∈ N & x d y-y d x+x y(x d y+y d x)=0 & (d y)/(y)-(d x)/(x)+(x d y+y d x)=0 & n|y|- n|x|+x y=c & y(1)=2 & n|2|-0+2=c & c=2+ n 2 & n|y|- n|x|+x y=2+ n 2 & n|x|= n|(y)/(2)|-2+x y & |x|=|(y)/(2)| e^ x y-2 & 2|x|=|y| e^ x y-2 & α=2 β=2 α+β=4 aligned

JEE Main 20248 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 8 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2024 · 8 April · Evening Shift · Mathematics · Differential Equations

Let α|x|=|y| e^ x y-β , α, β ∈ N be the solution of the...

Let

\alpha|x|=|y| \mathrm{e}^{x y-\beta}, \alpha, \beta \in \mathbf{N}

be the solution of the differential equation

x \mathrm{~d} y-y \mathrm{~d} x+x y(x \mathrm{~d} y+y \mathrm{~d} x)=0

y(1)=2

. Then

\alpha+\beta

is equal to ________

Correct answer

Step-by-step solution

\begin{aligned} & a|x|=|y| e^{y x-\beta}, a, b \in N \\ & x d y-y d x+x y(x d y+y d x)=0 \\ & \frac{d y}{y}-\frac{d x}{x}+(x d y+y d x)=0 \\ & \ell n|y|-\ell n|x|+x y=c \\ & y(1)=2 \\ & \ell n|2|-0+2=c \\ & c=2+\ell n 2 \\ & \ell n|y|-\ell n|x|+x y=2+\ell n 2 \\ & \ell n|x|=\ell n\left|\frac{y}{2}\right|-2+x y \\ & |x|=\left|\frac{y}{2}\right| e^{x y-2} \\ & 2|x|=|y| e^{x y-2} \\ & \alpha=2 \quad \beta=2 \quad \alpha+\beta=4 \end{aligned}

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