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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

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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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The given equation is

straight y squared minus 2 ay plus straight x squared space equals space straight a squared

...(1)
Differentiating both sides w.r.t.x, we get,

2 space straight y space dy over dx minus 2 space straight a dy over dx plus 2 straight x equals 0 space space space or space space straight y dy over dx plus straight x space equals space straight a dy over dx

therefore space space space straight a space equals space fraction numerator straight y begin display style dy over dx end style plus straight x over denominator begin display style dy over dx end style end fraction

...(2)
Eliminating m from (1) and (2), we get,

straight y squared minus fraction numerator 2 straight y squared begin display style dy over dx end style plus 2 xy over denominator begin display style dy over dx end style end fraction plus straight x squared space equals space open parentheses straight y begin display style dy over dx end style plus straight x close parentheses squared over open parentheses begin display style dy over dx end style close parentheses squared

open parentheses straight x squared plus straight y squared close parentheses open parentheses dy over dx close parentheses squared minus 2 straight y squared open parentheses dy over dx close parentheses squared minus 2 xy open parentheses dy over dx close parentheses space equals space open parentheses straight y dy over dx plus straight x close parentheses squared

open parentheses straight x squared plus straight y squared close parentheses open parentheses dy over dx close parentheses squared minus 2 straight x squared open parentheses dy over dx close parentheses squared minus 2 xy dy over dx equals straight y squared open parentheses dy over dx close parentheses squared plus straight x squared plus 2 xy dy over dx

open parentheses straight x squared minus 2 straight y squared close parentheses space open parentheses dy over dx close parentheses squared space minus space 4 xy dy over dx minus straight x squared space equals space 0

left parenthesis 2 straight y squared minus straight x squared right parenthesis space open parentheses dy over dx close parentheses squared plus 4 xy dy over dx plus straight x squared space equals space 0

which is required differential equation.

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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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