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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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The given differential equation is

straight x squared plus straight y squared space equals space ax cubed

...(1)

2 straight x plus 2 straight y dy over dx space equals 3 ax squared

...(2)
Multiply (1) by 3 and (2) by x, we get,

3 straight x squared plus 3 straight y squared space equals space 3 ax cubed

...(3)

2 straight x squared plus 2 xy dy over dx space equals space 3 ax cubed

...(4)
Subtracting (4) from (3), we get

straight x cubed plus 3 straight y squared minus 2 xy dy over dx equals 0

straight x squared plus 3 straight y squared equals 2 xy dy over dx

which is required differential equation.

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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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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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