The solution of $(2 x + 3 y - 5) d x + (3 x - 4 y + 1) d y = 0$ is:

x^{2} + 3 x y - 2 y^{2} - 5 x + y = c

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x^{2} + 3 x y - 4 y^{2} - 5 x - y = c

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x^{2} + 3 y^{2} - 2 x y + 5 x + y = c

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x^{2} - 3 y^{2} - 2 x y + 5 x + y = c

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Solution

The correct option is B $x^{2} + 3 x y - 2 y^{2} - 5 x + y = c$
$\Rightarrow \frac{d Y}{d X} + \frac{(2 X + 3 Y - 5)}{(3 X - 4 X + 1)} = 0$

Put $x = X + h; y = Y + k$

$\Rightarrow 2 h + 3 k - 5 = 0................1$

$3 h - 4 k + 1 = 0..........2$

$\Rightarrow \frac{d Y}{d X} + \frac{2 X + 3 Y}{3 X - 4 Y} = 0$

put $Y = v X$

$\Rightarrow v + X \frac{d v}{d X} + \frac{2 + 3 v}{3 - 4 v} = 0$

$\Rightarrow X \frac{d v}{d X} + \frac{2 + 6 v - 4 v^{2}}{3 - 4 v} = 0$

$\Rightarrow \int (\frac{3 - 4 v}{1 + 3 v - 2 v^{2}}) d v + \int 2 \frac{d X}{X} = log c$

$\Rightarrow log X^{2} (1 + 3 v - 2 v^{2}) = log c$

$X^{2} + 3 X Y - 2 Y^{2} = c$

$\Rightarrow {(x - h)}^{2} + 3 (x - h) (y - k) - 2 {(y - k)}^{2} = c$

$\Rightarrow x^{2} - 2 y^{2} + 3 x y + x (- 2 h - 3 k) + y (4 k - 3 h) = c$

from 1 and 2
$x^{2} - 2 y^{2} + 3 x y - 5 x + y = c$

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Q. If A =

2 x^{2} - y + 3 x y

, B =

x^{2} + 2 x y

and C = 3y + 2xy, then find A - B + C.

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x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 2 = 0

and

x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + c = 0

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Q. Classify the following polynomials as monomials, binomials, and trinomials. Which polynomial do not fit in any of these three categories?

x + y, 1000, x + x^{2} + x^{3} + x^{4}, 7 + y + 5 x, 2 y - 3 y^{2}, 2 y - 3 y^{2} + 4 y^{3},

5 x - 4 y + 3 x y, 4 z - 15 z^{2}, a b + b c + c d + d a, p q r, p^{2} q + p q^{2}, 2 p + 2 q

Question 2
Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories:
$x + y, 1000, x + x^{2} + x^{3} + x^{4}, 7 + y + 5 x, 2 y - 3 y^{2},$ $2 y - 3 y^{2} + 4 y^{3}, 5 x - 4 y + 3 x y, 4 z - 15 z^{2},$
$a b + b c + c d + d a, p q r, p^{2} q + p q^{2}, 2 p + 2 q$

Q. From the sum of x² + 3y² − 6xy, 2x² − y² + 8xy, y² + 8 and x² − 3xy subtract −3x² + 4y² − xy + x − y + 3.

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The solution of (2x+3y−5)dx+(3x−4y+1)dy=0 is:

The solution of $(2 x + 3 y - 5) d x + (3 x - 4 y + 1) d y = 0$ is: