Solutions to Simpler Differential Equations

(11)   Ö(2 xy) (dy/dx) = 1

 Rewrite:

Ö(y)  dy =   dx/ Ö(2 x)

Integrate:

ò y ^1/2 dy =   ò dx/ Ö(2 x)

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

1²  +  (2) ²   = 5

So the particular solution is:

  x²    +   y ²     = 5