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Substitution Method

The substitut...

Question

The substitution required to change $(3 y - 7 x + 7) d x + (7 y - 3 x + 3) d y = 0$ to a homogeneous differential equation is

x = X + 1, y = Y

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x = X, y = Y + 1

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x = X + 1, y = Y + 1

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x = X + 2, y = Y + 2

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Solution

The correct option is B $x = X + 1, y = Y$
Let x $=$ X+h and y $=$ Y+K
$3 (Y + K) - 7 (X + h) + 7 = 3 Y - 7 X$
$\Rightarrow 3 K - 7 h + 7 = 0$
$\Rightarrow 7 h - 3 K = 7$ -------------(1)
$7 (Y + K) - 3 (X + h) + 3 = - 3 X + 7 Y$
$\Rightarrow 3 = 3 h - 7 K$ -----------------(2)
$(7 h - 3 k = 7) \times 3$
$- (3 h - 7 k = 3) \times 7$
------------------------------------------
$\Rightarrow 40 k = 0 \Rightarrow k = 0$
------------------------------------------
$\Rightarrow h = 1$
$\Rightarrow x = X + 1; y = Y + 0 = Y$

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