## Lakshya Education MCQs

Question: If xdydx=y(log ylog x+1), then the solution of the equation is
Options:
 A. y log(xy)=cx B. x log(yx)=cy C. log(yx)=cx D. log(xy)=cx
: C

dydx=yx(logyx+1)
Put y=vxdydx=v+xdvdx
v+xdvdx=vlogv+v1vlogvdv=1xdx1vlogvdv=1xdxlog(logv)=logx+logc
logyx=cx

Earn Reward Points by submitting Detailed Explaination for this Question

## More Questions on This Topic :

Question 1. The degree and order of the differential equation of the family of all parabolas whose axis is x–axis, are respectively
1.    1,2
2.    3,2
3.    2,3
4.    2,1
: A

Equation of family of parabolas with x-axis as axis is y2=4a(x+α) where a,α are two arbitrary constants. So differential equation is of order 2 and degree 1.
Question 2. The solution of dydx=yx+tanyx is
1.    y sin(yx)=cx
2.    y sin(yx)=cy
3.    sin(xy)=cx
4.    sin(yx)=cx
: D

Put y=vx.Thendydx=v+xdvdx
Given equation is dydx=vx+tanyxv+xtanvcotvdv=dxxlogsinv=logx+logc
sinv=cxsin(yx=cx)
Question 3. If y1(x) is a solution of the differential equation dydxf(x)y=0, then a solution of the differential equation dydx+f(x)y=r(x)
1.    1y1(x)∫r(x)y1(x)dx
2.    y1(x)∫r(x)y1(x)dx
3.    ∫r(x)y1(x)dx
4.    None of these
: A

dydxf(x).y=0dyy=f(x)dx
ln y=f(x)dx
y1(x)=ef(x)dxThen for given equation I.F = ef(x)dx
Hence Solution y.y1(x)=r(x).y1(x)dx
y=1y1(x)r(x).y1(x)dx
Question 4. The solution of the differential equation (x2sin3yy2cosx)dx+(x3cosysin2y2ysinx)dy=0 is
1.    x3sin3y=3y2sinx+C
2.    x3sin3y+3y2sinx=C
3.    x2sin3y+y3sinx=C
4.    2x2siny+y2sinx=C
: A

(x2sin3yy2cosx)dx+(x3cosysin2y2ysinx)dy=0dydx=y2cosxx2sin3yx3cosysin22ysinx(x3cosysin2y2ysinx)dy=(y2cosxx2sin3y)dx=0(x33dsin3ysindy2)sin3yd(x33)+y2dsinx=0
x33dsin2y+sin3yd(x33)(sindy2+y2dsinx)
d(x33sin3y)d(y2sinx)=0x33sin3yy2sinx=c
Question 5. The order of the differential equation whose general solution is given by y=C1e2x+C2+C3ex+C4sin(x+C5) is
1.    5
2.    4
3.    3
4.    2
: B

y=C1e2x+C2+C3ex+C4sin(x+C5)=C1.eC2e2x+C3ex+C4(sinxcosC5+cosxsinC5)=Ae2x+C3ex+Bsinx+Dcosx
Here, A=C1eC2,B=C4cosC5,D=C4sinC5
(Since equation consists of four arbitrary constants)
order of differential equation = 4.
Question 6. The orthogonal trajectories of the family of curves an1y=xn are given by
1.    xn+n2y= constant
2.    ny2+x2= constant
3.    n2x+yn= constant
4.    n2x−yn= constant