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Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Page 2
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Alternatively: We can also use equation of circle ?? 2
+(?? -1)
2
+?? (?? -1)=0 and
proceed.
1.2 Find the curve for which the part of the tangent cut-off by the axes is bisected
at the point of tangency.
(2014 : 10 Marks)
Solution:
Let equation of tangent line at point ' ?? ' al
?? -?? ?? -?? =
????
????
(??)
Now, its point of intersection with co-ordinate axes are
?? (0,?? -
?? ????
????
);?? (?? -?? ????
????
,0)
Given : ' ?? ' is mid point of ???? .
So,
Page 3
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Alternatively: We can also use equation of circle ?? 2
+(?? -1)
2
+?? (?? -1)=0 and
proceed.
1.2 Find the curve for which the part of the tangent cut-off by the axes is bisected
at the point of tangency.
(2014 : 10 Marks)
Solution:
Let equation of tangent line at point ' ?? ' al
?? -?? ?? -?? =
????
????
(??)
Now, its point of intersection with co-ordinate axes are
?? (0,?? -
?? ????
????
);?? (?? -?? ????
????
,0)
Given : ' ?? ' is mid point of ???? .
So,
?? -?? ????
????
2
=?? and
?? -
?? ????
2
=??
? ?? =-?? ????
????
and ?? =-
?? ????
????
?
????
????
=-
?? ??
? ?????? +?????? =0
Integrating, we get
?? 2
+?? 2
=?? which the required curve.
1.3 Find the differential equation (DE) representing all the circles in the ???? -plane.
(2017 : 10 Marks)
Solution:
Method 1: General equation of circle
(?? -?? )
2
+(?? -?? )
2
=?? 2
Differentiating w.r.t. ?? ,
2(?? -?? )+2(?? -?? )
????
????
=0
??.?? ., (?? -?? )+(?? -?? )?? 1
=0 (??)
Differentiating again w.r.t. ??
1+(?? -?? )?? 2
+?? 1
2
=0 (???? )
Differentiating again w.r.t. ??
(?? -?? )?? 3
+?? 1
?? 2
+2?? 1
?? 2
=0
i.e.,
(?? -?? )=
-3?? 1
?? 2
?? 3
Substituting it in (ii)
Page 4
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Alternatively: We can also use equation of circle ?? 2
+(?? -1)
2
+?? (?? -1)=0 and
proceed.
1.2 Find the curve for which the part of the tangent cut-off by the axes is bisected
at the point of tangency.
(2014 : 10 Marks)
Solution:
Let equation of tangent line at point ' ?? ' al
?? -?? ?? -?? =
????
????
(??)
Now, its point of intersection with co-ordinate axes are
?? (0,?? -
?? ????
????
);?? (?? -?? ????
????
,0)
Given : ' ?? ' is mid point of ???? .
So,
?? -?? ????
????
2
=?? and
?? -
?? ????
2
=??
? ?? =-?? ????
????
and ?? =-
?? ????
????
?
????
????
=-
?? ??
? ?????? +?????? =0
Integrating, we get
?? 2
+?? 2
=?? which the required curve.
1.3 Find the differential equation (DE) representing all the circles in the ???? -plane.
(2017 : 10 Marks)
Solution:
Method 1: General equation of circle
(?? -?? )
2
+(?? -?? )
2
=?? 2
Differentiating w.r.t. ?? ,
2(?? -?? )+2(?? -?? )
????
????
=0
??.?? ., (?? -?? )+(?? -?? )?? 1
=0 (??)
Differentiating again w.r.t. ??
1+(?? -?? )?? 2
+?? 1
2
=0 (???? )
Differentiating again w.r.t. ??
(?? -?? )?? 3
+?? 1
?? 2
+2?? 1
?? 2
=0
i.e.,
(?? -?? )=
-3?? 1
?? 2
?? 3
Substituting it in (ii)
1+(
-3?? 1
?? 2
?? 3
)?? 2
+?? 1
2
=0
??.?? ., (1+?? 1
2
)?? 3
-3?? 1
?? 2
2
=0
??.?? ., (1+?? 1
2
)?? 3
=3?? 1
?? 2
2
Method II : Using curvature-formula (?? ) .
1.4 Find the orthogonal trajectories of the family of circles passing the points
(?? ,?? ) and (?? ,-?? ) .
(2020: 10 marks)
Solution:
Let equation of circle through (0,2) and (0,-2) be:
?? 2
+(?? 2
-4)+???? =0, ?? : Parameter (1)
Differentiating w.r.t. ?? , we get:
2?? +2?? (
????
????
)+?? =0 (2)
From (1) and (2),
?? 2
+?? 2
-4+[-2?? -2?? (
????
????
)]?? =0
?? 2
-?? 2
-4-2????
????
????
=0 (3)
Replace
????
????
by -
????
????
in (3), we get
?? 2
-?? 2
-4+2????
????
????
=0
(?? 2
-4)????
?? 2
+
2???????? -?? 2
????
?? 2
=0
? (1-
4
?? 2
)???? +? ?? (
?? 2
?? ) =0
?? +
4
?? +
?? 2
?? =0
?? 2
+?? 2
+4 =???? (required trajectory)
1.5 Find the orthogonal trajectories of the family of confocal conics
?? ?? ?? ?? +?? +
?? ?? ?? ?? +?? =?? ;?? >?? >??
Page 5
Edurev123
Ordinary Differential Equations
1. Formulation of Differential Equations
1.1 Find the differential equations of the family of circles in the ???? -plane passing
through (-?? ,?? ) and (?? ,?? ) .
(2009 : 20 Marks)
Solution:
Approach : First use conditions to get the general equation of such a circle. Then get the
differential equations
General equation of circle in ???? plane is
?? 2
+?? 2
+2???? +2???? +?? =0 (??)
It passes through (-1,1) and (1,1)
? 2-2?? +2?? +?? =0?4?? =0
2+2?? +2?? +?? =0??? =-(2?? +2)
? General equation of circles passing through (-1,1) and (1,1) is
?? 2
+?? 2
+2???? -(2?? +2)=0 (???? )
where ?? is the single parameter.
Differentiating (ii) with respect to ??
2?? +2?? ????
????
+2?? ????
????
=0
?
????
????
=
-?? ?? +?? ??? =
-?? ???? /????
-??
Putting this in (ii)
?? 2
+?? 2
+2(
-?? ???? /????
-?? )?? -[2(
-?? ???? /????
-?? )+2]=0
? ?? 2
-?? 2
-
2????
???? /????
+
2?? ???? /????
+2?? -2=0
?
????
????
=
?? 2
-?? 2
-2?? +2
2?? (1-?? )
which is the required differential equation.
Alternatively: We can also use equation of circle ?? 2
+(?? -1)
2
+?? (?? -1)=0 and
proceed.
1.2 Find the curve for which the part of the tangent cut-off by the axes is bisected
at the point of tangency.
(2014 : 10 Marks)
Solution:
Let equation of tangent line at point ' ?? ' al
?? -?? ?? -?? =
????
????
(??)
Now, its point of intersection with co-ordinate axes are
?? (0,?? -
?? ????
????
);?? (?? -?? ????
????
,0)
Given : ' ?? ' is mid point of ???? .
So,
?? -?? ????
????
2
=?? and
?? -
?? ????
2
=??
? ?? =-?? ????
????
and ?? =-
?? ????
????
?
????
????
=-
?? ??
? ?????? +?????? =0
Integrating, we get
?? 2
+?? 2
=?? which the required curve.
1.3 Find the differential equation (DE) representing all the circles in the ???? -plane.
(2017 : 10 Marks)
Solution:
Method 1: General equation of circle
(?? -?? )
2
+(?? -?? )
2
=?? 2
Differentiating w.r.t. ?? ,
2(?? -?? )+2(?? -?? )
????
????
=0
??.?? ., (?? -?? )+(?? -?? )?? 1
=0 (??)
Differentiating again w.r.t. ??
1+(?? -?? )?? 2
+?? 1
2
=0 (???? )
Differentiating again w.r.t. ??
(?? -?? )?? 3
+?? 1
?? 2
+2?? 1
?? 2
=0
i.e.,
(?? -?? )=
-3?? 1
?? 2
?? 3
Substituting it in (ii)
1+(
-3?? 1
?? 2
?? 3
)?? 2
+?? 1
2
=0
??.?? ., (1+?? 1
2
)?? 3
-3?? 1
?? 2
2
=0
??.?? ., (1+?? 1
2
)?? 3
=3?? 1
?? 2
2
Method II : Using curvature-formula (?? ) .
1.4 Find the orthogonal trajectories of the family of circles passing the points
(?? ,?? ) and (?? ,-?? ) .
(2020: 10 marks)
Solution:
Let equation of circle through (0,2) and (0,-2) be:
?? 2
+(?? 2
-4)+???? =0, ?? : Parameter (1)
Differentiating w.r.t. ?? , we get:
2?? +2?? (
????
????
)+?? =0 (2)
From (1) and (2),
?? 2
+?? 2
-4+[-2?? -2?? (
????
????
)]?? =0
?? 2
-?? 2
-4-2????
????
????
=0 (3)
Replace
????
????
by -
????
????
in (3), we get
?? 2
-?? 2
-4+2????
????
????
=0
(?? 2
-4)????
?? 2
+
2???????? -?? 2
????
?? 2
=0
? (1-
4
?? 2
)???? +? ?? (
?? 2
?? ) =0
?? +
4
?? +
?? 2
?? =0
?? 2
+?? 2
+4 =???? (required trajectory)
1.5 Find the orthogonal trajectories of the family of confocal conics
?? ?? ?? ?? +?? +
?? ?? ?? ?? +?? =?? ;?? >?? >??
are constants and ?? is a parameter. Show that the given family of curves is self
orthogonal.
[2021 : 10 marks]
Solution:
(i)
Given:
?? 2
?? 2
+?? +
?? 2
?? 2
+?? =1
Differentiating (i)
2?? ?? 2
+?? +
2?? ?? 2
+?? ????
????
=0
or ?? (?? 2
+?? )+?? (?? 2
+?? )
????
????
=0
or,
?? (?? +?? ????
????
)=-(?? 2
?? +?? 2
?? ????
????
)
? ?? =-[
?? 2
?? +?? 2
?? (
????
????
)
?? +?? (
????
????
)
]
? ?? 2
+?? =?? 2
-
?? 2
?? +?? 2
?? (
????
????
)
?? +?? (
????
????
)
=
(?? 2
-?? 2
)?? ?? +?? (
?? ?? ?? ????
)
and
?? 2
+?? =?? 2
-
?? 2
?? +?? 2
?? (
????
????
)
?? +?? (
????
????
)
=
-(?? 2
-?? 2
)?? (
????
????
)
?? +?? (
????
????
)
Putting the above values of (?? 2
+?? ) and (?? 2
+?? ) in (i),
We have,
?? 2
[?? +?? (
????
????
)]
(?? 2
-?? 2
)?? -
?? 2
[?? +?? (
????
????
)]
(?? 2
-?? 2
)?? (
????
????
)
=1
or [?? +?? (
????
????
)][?? -?? (
?? ?? ????
)]=?? 2
-??
which is the differential equation of the family (i). Replacing ???? /???? by ( -?????????? ) in (ii),
the differential equation of the required orthogonal trajectories is
or [?? +?? (-
????
????
)][?? -?? (-
????
????
)]=?? 2
-?? 2
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Dec 22, 2024 Last updated |
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