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Differential Equations

Short Answer Type

51. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1

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52. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

straight x squared over straight a squared minus straight y squared over straight b squared space equals space 1

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53. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

straight y squared minus 4 straight a left parenthesis straight x minus straight b right parenthesis

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54. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

left parenthesis straight y minus straight b right parenthesis squared space equals space 4 left parenthesis straight x minus straight a right parenthesis

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55. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).
$straight x squared plus straight y squared space equals space 2 ax$

The given equation is
$straight x squared plus straight y squared space equals space 2 ax$ ...(1)
Differentiating both sides w.r.t. x, we get,
$2 straight x plus 2 straight y dy over dx space equals space 2 straight a$ ...(2)
Putting value of 2a from (2) in (1), we get,
$straight x squared plus straight y squared space equals space open parentheses 2 straight x plus 2 straight y dy over dx close parentheses straight x$
or $straight x squared plus straight y squared space equals space 2 straight x squared plus 2 xy dy over dx space space space space or space space space 2 xy dy over dx plus straight x squared minus straight y squared space equals space 0$
which is the required differential equation.

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56.

Find the differential equation that will represent the family of curves given by (a, b: parameter):
y = ax³

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57.

Find the differential equation that will represent the family of curves given by (a, b: parameter):
x² + y² = a x³

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58.

Find the differential equation that will represent the family of curves given by (a, b: parameter):
y = e^ax

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Long Answer Type

59. Form the differential equation corresponding to y² – 2 a y + x² – a² by eliminating a.

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Short Answer Type

60. Form the differential equation representing the family of curves y = a sin (x + b), where a and b are arbitrary constants.

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