Show that y2-x2-x y-a is a solution of the differential equation x-2 y d y d x-2 x-y-0-

We have, x 2ydydx+2x+y=0 Now,y2 minus; x2 minus; xy = a #8658;2ydydx 2x y xdydx=0 #8658;2y xdydx 2x y=0 #8658;2y xdydx=2x+y #8658;x 2ydydx= 2x+y ...

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Standard XII

Mathematics

Ordinary Differential Equations

Show that y2-...

Question

Show that y² − x² − xy = a is a solution of the differential equation $(x - 2 y) \frac{d y}{d x} + 2 x + y = 0$ .

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Solution

We have,
$(x - 2 y) \frac{d y}{d x} + 2 x + y = 0$
Now,y² − x² − xy = a
^{$\Rightarrow 2 y \frac{d y}{d x} - 2 x - y - x \frac{d y}{d x} = 0 \Rightarrow (2 y - x) \frac{d y}{d x} - 2 x - y = 0 \Rightarrow (2 y - x) \frac{d y}{d x} = 2 x + y \Rightarrow (x - 2 y) \frac{d y}{d x} = - (2 x + y) \Rightarrow (x - 2 y) \frac{d y}{d x} + 2 x + y = 0$}
Thus, y² − x² − xy = a is the solution of the given differential equation.

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Show that y2 − x2 − xy = a is a solution of the differential equation x-2ydydx+2x+y=0.

We have, x-2ydydx+2x+y=0 Now,y2 − x2 − xy = a ⇒2ydydx-2x-y-xdydx=0⇒2y-xdydx-2x-y=0⇒2y-xdydx=2x+y⇒x-2ydydx=-2x+y⇒x-2ydydx+2x+y=0 Thus, y2 − x2 − xy = a is the solution of the given differential equation.

Show that y² − x² − xy = a is a solution of the differential equation $(x - 2 y) \frac{d y}{d x} + 2 x + y = 0$ .