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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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The given differential equation is

straight y equals straight e to the power of ax space space space space space space space space space space space space space space space space space space space space space space space space space space space or space space space space log space straight y space equals space log space straight e to the power of ax

therefore space space space space space space space log space straight y space space equals space straight a space straight x

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

1 over straight y dy over dx equals straight a

...(2)
From (1) and (2), we get,

log space straight y space equals space 1 over straight y dy over dx. straight x

straight x dy over dx space equals space straight y space log space straight y

which is required differential equation.

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

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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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