Home
Class 12
MATHS
If y(x) is the the solution of the diffe...

If y(x) is the the solution of the differntial equation `dy/dx+((2x+1)/x)y=e^(-2x),xgt0," where "y(1)=1/2e^(-2)`, then

A

(a) y(x) is decreasing in `(1/2,1)`

B

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

C

(c) `y(log_(e) 2) =log_(e)2`

D

(d) `y(log_(e) 2) =log_(e)2/4`

Text Solution

Verified by Experts

The correct Answer is:
(a)
We have , `dy/dx+((2x+1)/1)y=e^(-2x)`
which is of the form `dy/dx+Py=Q,` where
`P=(2x+1)/1 and Q=e^(-2x)`
Now, `IF=e^(intPdx)=e^(int((1=2x)/x)dx)=e^(int(1/x+2)dx`
`=e^(lnx+2x)=e^(lnx).e^(2x)=x.e^(2x)`
and the solution of the given equation is
`y cdot (IF) = int (IF) Q dx+C`
` rArr y(xe^(2x))=int (xe^(2x).e^(-2x))dx+C`
`=int xdx+C=x^(2) / 2 +C …(i) `
Since, `y=1/2e^(-2)` when x=1
`therefore 1/2e^(-2).e^(2)=1/2+C rArr C=0` (using Eq. (i))
`therefore y(xe^(2x))=x^(2)/2 rArr y=x/2e^(-2x)`
Now,`dy/dx =1/2e^(-2x)+x/2e^(-2x)(-2)=e^(-2x){1/2-x}lt0,`
`if1/2ltxlt1` "[by using product rule of derivative ]"
and `y(log_(e)2) =(log_(e)2)/2e^(_2log_(e)2)=1/2 log_(e)2e^(log_(e)2^(-2)`
`=1/2. log_(e)2cdot2^(-2)=1/8log_(e)2`
Promotional Banner

Similar Questions

Explore conceptually related problems

Solution of the differential equation dy/dx=(y^3)/(e^(2x)+y^2) , is

The solution of the differential equation (dy)/(dx)=1/(x y[x^2siny^2+1]) is

The solution of the differential equation (dy)/(dx)=(3x^2y^4+2x y)/(x^2-2x^3y^3) is

The solution of the differential equation x(x^2+1)((dy)/(dx))=y(1-x^2)+x^3logx is

The solution of differential equation (2y+x y^3)dx+(x+x^2y^2)dy=0 is

The general solution of the differential equation e^(x) dy + (y e^(x) + 2x)dx = 0 is

Let y = y(x) be the solution of the differential equation x dy/dx+y=xlog_ex,(xgt1)." If " 2y(2)=log_e4-1," then "y(e) is equal to

The solution of the differential equation (d^2y)/(dx^2)=sin3x+e^x+x^2 when y_1(0)=1 and y(0)=0 is

The solution of the differential equation x^2(dy)/(dx)cos(1/x)-ysin(1/x)=-1, where y->-1 as x->oo is

Find the particular solution of the differential equation (dy)/(dx)=-4xy^(2), given that y=-1, where x=0