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Home  >>  CBSE XII  >>  Math  >>  Differential Equations
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Form the differential equation of the family of ellipses having foci on $y$-axis and centre at origin.

$\begin{array}{1 1}(A)xyy'' -x(y')^2 - yy' = 0 \\(B)\;xyy'' + x(y')^2 - yy' = 0 \\(C)\;xyy'' + x(y')^2 + yy' = 0 \\(D)\;xyy'' + x(y')^2 +y' = 0 \end{array} $

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1 Answer

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Toolbox:
  • It is given that the ellipse is having foci on y-axis and the centre at the origin
  • Hence the equation of the ellipse is $\large\frac{x^2}{b^2} +\frac{ y^2}{a^2 }$$= 1$
Step 1:
Using the information from the tool box we write the equation as
$\large\frac{x^2}{b^2} +\frac{ y^2}{a^2}$$ = 1$--------(1)
Differentiating this on both sides we get,
$\large\frac{2x}{b^2} + \frac{2yy'}{b^2}$$ = 0$--------(2)
Step 2:
Again differentiating this we get,
$\large\frac{1}{b^2} + (\frac{1}{a^2})$$[yy'' +(y')^2] = 0$
$\large\frac{1}{b^2} = - [\frac{1}{a^2}]$$(yy'' + (y')^2 $
Substituting this in equation (2) we get,
$-[\large\frac{x}{a^2}]$$[yy'' + (y')^2] +\large\frac{ yy'}{a^2}$$= 0$
On simplifying we get
$xyy'' + x(y')^2 - yy' = 0$
answered Aug 15, 2013 by sreemathi.v
 

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