wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

A solution of the differential equation (x2+xy+4x+2y+4)dydxy2=0, x>0, passes through the point (1,3). Then the solution curve:

A
intersects y=x+2 exactly at one point
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
intersects y=x+2 exactly at two points
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
intersects y=(x+2)2
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
does NOT intersect y=(x+3)2
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
Open in App
Solution

The correct option is D does NOT intersect y=(x+3)2
(x2+xy+4x+2y+4)dydxy2=0, x>0
(x+2)(x+y+2)dydx=y2

Let x+2=t
t(t+y)dydt=y2
dydt=(y/t)21+y/t

Let yt=k
k+tdkdt=k21+k
(k+1k)dk+dtt=0
k+ln|kt|=c
yx+2+ln|y|=c

Now, Given that the curve passes through the point (1,3).
c=ln3e
yx+2+ln|y|=ln3e

Option 1 & 2:
For the curve y=x+2
1+ln(x+2)=ln3e
LHS is an increasing function & RHS is a constant
Min(1+ln(x+2)=ln2e<ln3e
So, there will be only one intersection point of the curves

Option 3:
For the curve y=(x+2)2
x+2+2ln(x+2)=ln3e
LHS is an increasing function & RHS is a constant
Min(x+2+2ln(x+2))=2+2ln2>ln3e
So, the curve won't intersect eachother

Option 4:
For the curve y=(x+2)3
(x+2)2+3ln(x+2)=ln3e
LHS is an increasing function & RHS is a constant
Min((x+2)2+3ln(x+2)=4+3ln2>ln3e
So, the curve won't intersect eachother

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Methods of Solving First Order, First Degree Differential Equations
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon