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In the first column solutions of the differential equations are given and in second column differential equations after elimination of arbitrary constants are given
Column – I Column – II
A)
$y = c_{1} e^{2 x} + c_{2} e^{3 x} + c_{3} e^{- x}$
$(c_{1}, c_{2}, c_{3} arbitary)$
P) $\frac{d^{3} y}{d x^{3}} - 6 \frac{d^{2} y}{d x^{2}} + 12 \frac{d y}{d x} - 8 y = 0$
B)
$y = (c_{1} + c_{2} x + c_{3} x^{2}) e^{2 x}$
$(c_{1}, c_{2}, c_{3} arbitary)$
Q) $x [y \frac{d^{2} y}{d x^{2}} + {(\frac{d y}{d x^{}})}^{2}] = y \frac{d y}{d x}$
C) $x y = a e^{x} + b e^{- x} + x^{2} (a, b arbitary)$ R) $\frac{d^{3} y}{d x^{3}} - 4 \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} + 6 y = 0$
D) $a x^{2} + b y^{2} = 1 (a, b arbitary)$ S) $x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$

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$A - Q; B - P; C - S; D - R$

$A - R; B - Q; C - P; D - S$

$A - R; B - P; C - S; D - Q$

$A - Q; B - P; C - R; D - Q$

answer is A.

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Detailed Solution

$A) y = c_{1} e^{α x} + c_{2} e^{β x} + c_{3} e^{γ x} y_{3} - (\sum α) y_{2} + (\sum α β) y_{1} - (α β γ) y = 0$

$B) y = (c_{1} + c_{2} x + c_{3} x^{2}) e^{α x} y_{3} - 3 α y_{2} + 3 α^{2} y_{1} - α^{3} y = 0$

$C) x y = a e^{x} + b e^{- x} + x^{2} x y_{1} + y = a e^{x} - b e^{- x} + 2 x x y_{2} + 2 y_{1} = x y - x^{2} + 2$

$D) a x^{2} + b y^{2} = 1 2 a x + 2 b y y_{1} = 0 \frac{y y_{1}}{x} = - \frac{a}{b}$

Differenciate w.r.t to $x$

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