Solve the following differential equation- xcosyxy dx-xdy-ysinyxxdy - ydx-

Divide by x ^ 2 ycos ( x y ) , we get (ydx+xdy) xy =tan ( y x ) ( xdy ydx x ^ 2 ) d(xy) xy =tan ( y x ) d( y x ) Integrating ,we g ...

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Standard XII

Mathematics

Derivative of Standard Functions

Solve the fol...

Question

Solve the following differential equation: $x cos (\frac{y}{x}) (y d x + x d y) = y s i n (\frac{y}{x}) (x d y - y d x) .$

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Solution

Divide by $x^{2} y cos (\frac{x}{y})$ , we get
$\frac{(y d x + x d y)}{x y} = tan (\frac{y}{x}) (\frac{x d y - y d x}{x^{2}}) \frac{d (x y)}{x y} = tan (\frac{y}{x}) d (\frac{y}{x})$
Integrating ,we get
$log (x y) = log sec (\frac{y}{x}) + log C$
or $\frac{x y}{sec (\frac{y}{x})} = C$
or $x y (cos (\frac{y}{x})) = C$

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Similar questions

Q. Solve:

x cos \frac{y}{x} (y d x + x d y) = y sin \frac{y}{x} (x d y - y d x)

Q. Find the solution of

x cos \frac{y}{x} (y d x + x d y) = y sin \frac{y}{x} (x d y - y d x)

Show that the given differential equation is homogeneous and then solve it.

${x c o s (\frac{y}{x}) + y s i n (\frac{y}{x})} y d x = {y s i n (\frac{y}{x}) - x c o s (\frac{y}{x})} x d y$

Q. If

x cos (\frac{y}{x}) (y d x + x d y) = y sin (\frac{y}{x}) (x d y - y d x), y (1) = 2 π

, then the value of

4 \frac{y (4)}{π} cos (\frac{y (4)}{4})

Q. In the following, show that the given differential equation is homogeneous and solve each of them.
1.

(x^{2} + x y) d y = (x^{2} + y^{2}) d x

y^{'} = \frac{x + y}{x}

(x y) d y (x + y) d x = 0

(x^{2} + y^{2}) d x + 2 x y d y = 0

x^{2} \frac{d y}{d x} = x^{2} - 2 y^{2} + x y

x d y . y d x = \sqrt{x^{2} + y^{2}} d x

{x cos (\frac{y}{x}) + y sin (\frac{y}{x})} y d x = {y sin (\frac{y}{x}) - x cos (\frac{y}{x})} x d y

x \frac{d y}{d x} - y + x sin (\frac{y}{x}) = 0

y d x + x log (\frac{y}{x}) d y - 2 x d y = 0

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Solve the following differential equation: xcos(yx)(ydx+xdy)=ysin(yx)(xdy−ydx).

Divide by x2ycos(xy), we get(ydx+xdy)xy=tan(yx)(xdy−ydxx2)d(xy)xy=tan(yx)d(yx)Integrating ,we getlog(xy)=logsec(yx)+logCor xysec(yx)=Cor xy(cos(yx))=C

Solve the following differential equation: $x cos (\frac{y}{x}) (y d x + x d y) = y s i n (\frac{y}{x}) (x d y - y d x) .$