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Question

Answer the following by appropriately matching the lists based on the information given in Column I and Column II
Column IColumn IIa. If the function y=e4x+2ex is a solution ofthe differential equation d3ydx313dydxy=K,then the value of K3 is p. 3b. Number of straight lines which satisfy the differential equation dydx+x(dydx)2y=0is q. 4c. If real value of m for which the substitution, y=um will transform the differential equat-ion, 2x4ydydx+y4=4x6 into a homogeneous equation, then the value of 2m is r. 2d. If the solution of differential equation x2d2ydx2+2xdydx=12y is y=Axm+Bxn,then |m+n| is s. 7

Then which of the following is correct ?

A
as, br, cp, dq
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B
aq, br, cp, ds
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C
aq, bp, cr, ds
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D
as, bq, cp, ds
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Solution

The correct option is B aq, br, cp, ds
a. y=e4x+2ex; y1=4e4x2ex; y2=16e4x+2ex; y3=64e4x2ex
Now, y313y1=(64e4x2ex)13(4e4x2ex)
=12e4x+24ex
y313y=12(e4x+2ex)=12y
K=12 and K3=4

b. Since equation is of degree 2, two lines are possible.

c. y=um or dydx=mum1dudx
Substituting the value of y and dydx in 2x4ydydx+y4=4x6,
we have
2x4ummum1dudx+u4m=4x6 or dudx=4x6u4m2mx4u2m1
For homogeneous equation, 4m=6 or m=32
and 2m1=2 or m=32.

d.
y=Axm+Bxn
dydx=Amxm1nBxn1
d2ydx2=Am(m1)xm2+n(n+1)Bxn2
Putting these values in x2d2ydx2+2xdydx=12y, we get
m(m+1)Axm+n(n1)Bxn=12(Axm+Bxn)
i.e., m(m+1)=12 or n(n1)=12
i.e., m=3,4 or n=4,3

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