Differential Equations 2 Question 10

10. If y(x) is the solution of the differential equation

dydx+2x+1xy=e2x,x>0

where y(1)=12e2, then

(2019 Main, 11 Jan I)

(a) y(x) is decreasing in 12,1

(b) y(x) is decreasing in (0,1)

(c) y(loge2)=loge4

(d) y(loge2)=loge24

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Answer:

Correct Answer: 10. (a)

Solution:

  1. We have, dydx+2x+1xy=e2x

which is of the form dydx+Py=Q, where

P=2x+1x and Q=e2x

Now, IF =ePdx=e1+2xxdx=e1x+2dx

=elnx+2x=elnxe2x=xe2x

and the solution of the given equation is

y(IF)=(IF)Qdx+Cy(xe2x)=(xe2xe2x)dx+C=xdx+C=x22+C

Since, y=12e2 when x=1

12e2e2=12+CC=0 (using Eq. (i))

y(xe2x)=x22y=x2e2x

Now, dydx=12e2x+x2e2x(2)=e2x12x<0,

if 12<x<1

[by using product rule of derivative]

and y(loge2)=loge22e2loge2=12loge2eloge22

=12loge222=18loge2



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