(11) Ö(2 xy) (dy/dx) = 1

Ö(y) dy
= dx/ Ö(2 x)

Integrate:

y ^{3/2} / 3/2 = Ö2 Ö(x)

2 y ^{3/2} = 3 (Ö2 ) Ö(x)

Then:

y ^{3/2} = 3 Ö x /2 + c

2) sin x (dx/dy) + cosh (2y)
= 1

ò
sin x dx + ò cosh (2y) dy = 1

ð - cos x + sinh
2y /2 = c

ð - 2 cos x + sinh
2y = c

3) ln x (dx/dy) = x/y

Separate variables:

ln x dx =
(x/y) dy

dy/y = ln x dx/x

Integrate:

ò dy/y
= ln x
ò dx/x

ln
y = ln x ln x + c

ln **| **y ** |****
** + c =
½ ln **| **x
^{2 }**|**

4) dy/dx = exp (x) (exp(-y)

=> dy/ exp (-y) = exp (x) dx

ò
exp (y) dy = ò exp (x) dx

exp (y) = exp (x) + C

5) Find the particular solution of:

x dx + y dy = 0; for y = 2 when x = 1

ò
x dx + ò
y dy = 0

x ^{2}/2^{ }+ y ^{2 }/2^{ }= c

or:

x^{ 2}
+ y ^{ 2} = r
^{ 2
= }2c

Where r ^{ 2
= }2c is the *constant
of integration*.

The solution then is the equation of a circle with the center at the origin and radius r.

For particular soln. use: y= 2, when x = 1

1^{2} + (2) ^{2} = 5

So the particular solution is:

x^{2 }+ y ^{2 }= 5

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