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1)

A solution curve of the differential  equation (x2+xy+4x+2y+4)dydxy2=0 , x>0, passes through the point (1,3) Then , the solution curve


A) intersects y=x+2 exactly at one point

B) intersects y=x+2 exactly at two points

C) intersects y=(x+2)2

D) does not intersect y=(x+3)2

Answer:

Option A,D

Explanation:

Given, (x2+xy+4x+2y+4)dydxy2=0

    [(x2+4x+4)+y(x+2)]dydxy2=0

    [(x+2)2+y(x+2)]dydxy2=0

Put x+2=X  and y=Y, then

(X2+XY)dYdXY2=0

     X2dY+XYdYY2dX=0

    X2dY+Y(XdYYdX)=0

    dYY=XdYYdXX2

   d(log|Y|)=d(YX)

On integrating both sides , we get,

 log|Y|=YX+C

where x+2=X and  y=Y

      log|y|=Yx+2+C              ...........(i)

 Since, it passes through the point (1,3)

        log3=1+C

  C=1log3

=(loge+log3)

 =log3e

  Eq. (i) becomes

 log|y|+yx+2log(3e)=0

        log(|y|3e)+yx+2=0       ........(ii)

Now, to check option (a)   y=x+2 intersects the curve.

      log(|x+2|3e)+x+2x+2=0

     log(|x+2|3e)=1

 |x+2|3e=e1=1e  

    |x+2|=3  or  x+2=±3

     x=1, -5 (rejected) , as x>0   [given]

    x=1 only one solution

 Thus ,(a)  is the correct  answer.

To check option (c) we have

 y=(x+2)2

and    log(|y|3e)+yx+2=0

  log[|x+2|23e]+(x+2)2x+2=0

   log[|x+2|23e]=(x+2)

     (x+2)23e=e(x+2)

or     (x+2)2.ex+2=3e

   ex+2=3e(x+2)2

2122021375_m18.JPG

Clearly, they have no solution.

 To check option (d),

y=(x+3)2

i.e,      log[|x+3|23e]+(x+3)2(x+2)=0

 To check the number of solutions,

  Let  g(x)=2log(x+3)+(x+3)2(x+2)log(3e)

      g(x)=2x+3+

((x+2).2(x+3)(x+3)2.1(x+2)2)0

 =2x+3+(x+3)(x+1)(x+2)2

Clearly, when x>0, then  ,g' (x)>0

    g(x) is increasing , when x>0

Thus ,when x.0, then g(x) >g(0)

g(x)>log(3e)+94>0

Hence, there is no solution

Thus, option  (d) is true