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

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The given equation is
y = a sin (x + b) ...(1)
Differentiating w.r.t.x, we get,

dy over dx space equals space straight a space cos left parenthesis straight x plus straight b right parenthesis

Again differentiating w.r.t.x, we get,

fraction numerator straight d squared straight y over denominator dx squared end fraction space equals negative asin left parenthesis straight x plus straight b right parenthesis

fraction numerator straight d squared straight y over denominator dx squared end fraction equals space minus straight y

open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight y space equals space 0

which is required differential equation.

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