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Solution of the differential equation (d...

Solution of the differential equation `(dy)/(dx)+(2y)/(x)=0`, where `y(1)=1`, is

A

a+b+c

B

`y(pi)=-3/5`

C

y(x) is decreasing in `(-pi/2,0)`

D

maximum value of y(x) is `1/sqrt5`

Text Solution

Verified by Experts

The correct Answer is:
D
It is a linear equation of the form `(dy)/(dx)+py=Q`
Where p=2, Q= cos x
Then I.F. = `e^(2x)`
Hence , the general solution is `y.e^(2x)=int e^(2x)cos x dx +c`
`y.e^(2x)=e^(2x)/5` (2 cos x + sin x ) +c
But, given that y(0)=`2/5 rArr ` c=0
So `y.e^(2x)=e^(2x)/5(2 cos x = sin x) rArr y=2/5 cos x + 1/5 sin x`
Now `y(pi/2)=1/5, y(pi)=(-2)/5, (dy)/(dx) > 0 AA x in (-pi/2, 0)`
So y(x) is increasing in `((-pi)/2,0)`
And maximum value of y(x) is `1/sqrt5`
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