Chapter 6 Differential Equations Ex 6.2

Question 1.
Obtain the differential equation by eliminating the arbitrary constants from the following equations:


Differentiating again w.r.t. x, we get
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Substituting the value of A in (1), we get
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This is the required D.E.

Alternative Method:
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Differentiating again w.r.t. x, we get,
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The equations (1), (2) and (3) are consistent in A and B.
∴ determinant of their consistency is zero.
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This is the required D.E.

(iii) y = A cos(log x) + B sin(log x)
Solution:

y = A cos(log x) + B sin (log x) ……. (1)
Differentiating w.r.t. x, we get
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Differentiating w.r.t. x, we get
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This is the required D.E.

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Differentiating twice w.r.t. x, we get
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This is the required D.E.

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Differentiating w.r.t. x, we get
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This is the required D.E.

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Differentiating w.r.t. x, we get
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Substituting the value of a in (1), we get
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This is the required D.E.

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The equations (1), (2) and (3) are consistent in c1e2x and c2e5x
∴ determinant of their consistency is zero.
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This is the required D.E.

Alternative Method:
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This is the required D.E.

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Differentiating w.r.t. x, we get
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Differentiating again w.r.t. x, we get
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The equations (1), (2) and (3) in c1, c2 are consistent.
∴ determinant of their consistency is zero.
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This is the required D.E.

(x) y = (A cos x + B sin x)
Solution:

y = (A cos x + B sin x)
∴  . y = A cos x + B sin x ………(1)
Differentiating w.r.t. x, we get
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Differentiating again w.r.t. x, we get
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This is the required D.E.

Question 2.
Form the differential equation of family of lines having intercepts a and b on the coordinate axes respectively.
Solution:

The equation of the line having intercepts a and b on the coordinate axes respectively, is
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where a and b are arbitrary constants.
[For different values of a and b, we get, different lines. Hence (1) is the equation of family of lines.]
Differentiating (1) w.r.t. x, we get
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Differentiating again w.r.t. x, we get d2ydx2=0
This is the required D.E.

Question 3.
Find the differential equation all parabolas having length of latus rectum 4a and axis is parallel to the X-axis.
Solution:

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Let A(h, k) be the vertex of the parabola whose length of latus rectum is 4a.
Then the equation of the parabola is (y – k = 4a (x – h), where h and k are arbitrary constants.
Differentiating w.r.t. x, we get
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Differentiating again w.r.t. x, we get
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This is the required D.E.

Question 4.
Find the differential equation of the ellipse whose major axis is twice its minor axis.
Solution:

Let 2a and 2b be lengths of major axis and minor axis of the ellipse.
Then 2a = 2(2b)
∴ a = 2b
∴ equation of the ellipse is
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This is the required D.E.

Question 5.
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
Solution:

The equation of the line parallel to the line 2x + 3y + 4 = 0 is 2x + 3y + c = 0, where c is an arbitrary constant.
Differentiating w.r.t. x, we get
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This is the required D.E.

Question 6.
Find the differential equation of all circles having radius 9 and centre at point (h, k).
Solution:

Equation of the circle having radius 9 and centre at point (h, k) is
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where h and k are arbitrary constant.
Differentiating (1) w.r.t. x, we get
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Differentiating again w.r.t. x, we get
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Substituting the value of (x – h) in (1), we get
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This is the required D.E.

Question 7.
Form the differential equation of all parabolas whose axis is the X-axis.
Solution:

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The equation of the parbola whose axis is the X-axis is
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Differentiating again w.r.t. x, we get
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This is the required D.E.