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Differential Equations

Short Answer Type

81. Form the differential equation of the family of circles having centre on x-axis and radius 3 units.

The equation of the family of circles having centre (0, b) on y-axis and radius 3 units is
x² + (y – b)² = 9    ...( 1)
Differentiating both sides w.r.t x, we get,
$2 straight x plus 2 left parenthesis straight y minus straight b right parenthesis dy over dx space equals space 0 space space space or space space space straight y space minus space straight b space equals space minus fraction numerator straight x over denominator begin display style dy over dx end style end fraction$
Putting this value of y - b in (1), we get,
   $straight x squared plus straight x squared over open parentheses begin display style dy over dx end style close parentheses squared space equals space 9$ or    $open parentheses straight x squared minus 9 close parentheses plus straight x squared over open parentheses begin display style dy over dx end style close parentheses squared space equals 0$

or    $left parenthesis straight x squared minus 9 right parenthesis space open parentheses dy over dx close parentheses squared plus straight x squared space equals space 0$

which is required differential equation.

124 Views

Long Answer Type

82. Find the differential equation of all the circles in the first quadrant which touch the co-ordinate axes.

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83. Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

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84. Form the differential equation of the family of circles touching the x-axis at origin.

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Short Answer Type

85. Form the differential equation of the family of circles touching the y-axis at origin.

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86. Form the differential equation of the family of parabolas having vertex at origin and axis along positive direction of x-axis.

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Long Answer Type

87. Form the differential equation of the family of parabolas having vertex at origin and axis along positive direction of y-axis.

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Short Answer Type

88. Form the differential equation representing the family of ellipses having foci on x-axis and centre at the origin.

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Long Answer Type

89. Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.

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90. Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

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