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

The equation of given curve is
$left parenthesis straight x minus straight b right parenthesis squared space space equals space 4 left parenthesis straight x minus straight a right parenthesis$ ...(1)
Differentiating w.r.t.x, we get,
$2 left parenthesis straight y minus straight b right parenthesis dy over dx space equals space 4$ or $left parenthesis straight y minus straight b right parenthesis space dy over dx equals 2$ ....(2)
Again differentiating both sides w.r.t x we get,
$left parenthesis straight y minus straight b right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx. dy over dx space space equals space 0 space space space space space or space space space fraction numerator 2 over denominator begin display style dy over dx end style end fraction fraction numerator straight d squared straight y over denominator dx squared end fraction plus open parentheses dy over dx close parentheses squared space equals space 0$ $open square brackets because space of space left parenthesis 2 right parenthesis close square brackets$
or $2 fraction numerator straight d squared straight y over denominator dx squared end fraction plus open parentheses dy over dx close parentheses cubed space equals space 0 comma$ which is required differential equation.

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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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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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