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The given differential equation is

straight x squared space equals space 2 straight y squared log space straight y

...(1)

therefore space space space space space

2 straight x space equals space 2 open curly brackets straight y squared cross times 1 over straight y dy over dx plus space log space straight y space open parentheses 2 space straight y space dy over dx close parentheses close curly brackets

2 straight x space equals 2 left parenthesis straight y plus 2 space straight y space log space straight y right parenthesis dy over dx

rightwards double arrow

straight x equals left parenthesis straight y plus 2 space straight y space log space straight y right parenthesis dy over dx

Multiplying both sides by y, we get

straight x space straight y space equals space left parenthesis straight y squared plus 2 straight y squared space log space straight y right parenthesis dy over dx

xy space equals space left parenthesis straight y squared plus straight x squared right parenthesis space dy over dx

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

left parenthesis straight x squared plus straight y squared right parenthesis space dy over dx minus straight x space straight y space equals space 0

Hence the result.

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42. The number of arbitrary constants in the general solution of a differential equation of fourth order are:

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43.

The number of arbitrary constants in the particular solution of a differential equation of third order are

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44. Find the differential equation that will represent the family of straight lines y = m x + c, where m, c (∈ R) are parameters.

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45.

Find the differential equations from y = k e^{sin^–1 x} + 3.

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46. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

straight x squared plus straight y squared space space equals space straight a squared

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47. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

straight x squared minus straight y squared space equals space straight a squared

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48. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

straight y squared equals 4 ax

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49. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

left parenthesis straight x minus straight a right parenthesis squared minus straight y squared space equals space 1

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50. Represent the following family of curves by forming the corresponding differential equations (a. b: parameters).

straight x over straight a plus straight y over straight b equals 1

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