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The differential equation representing the family of ellipses having foci either on the
x-axis or on the y-axis, centre at the origin and passing through the point (0, 3) is :

  • Option 1)

    xy y" + x(y')^{2}- yy'=0

  • Option 2)

    x+yy" = 0

  • Option 3)

    xy y' + y^{2}-9=0

  • Option 4)

    xy y' - y^{2}+9=0

 

Answers (2)

best_answer

As we have learned

Formation of Differential Equations -

A differential equation can be derived from its equation by the process of differentiation and other algebraical process of elimination

-

 

 

Formation of Differential Equations -

Let y and x be the dependent and the independent variables respectively. The equation of (x, y, c) = 0 , is family of curves

- wherein

C is the arbitrary Constant

 

 Equation of ellipse :

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1

pass through (0,3) 

\frac{9}{b^{2}}=1\Rightarrow b=\pm 3

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{9}=1  ; differentiating we get

\frac{2x}{a^{2}}+\frac{2y}{9}\frac{dy}{dx}=0

\frac{x}{a^{2}}= \frac{-y}{9}y'

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{9}=1

\frac{x}{a^{2}}x+\frac{y^{2}}{9}=1

(-xy)\frac{y'}{9}+\frac{y^{2}}{9}=1

-xyy'+y^{2}=9

xyy'-y^{2}+9=0

 

 

 

 

 

 

 

 

 


Option 1)

xy y" + x(y')^{2}- yy'=0

This is incorrect 

Option 2)

x+yy" = 0

This is incorrect 

Option 3)

xy y' + y^{2}-9=0

This is incorrect 

Option 4)

xy y' - y^{2}+9=0

This is correct 

Posted by

Himanshu

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