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41. For problem given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:

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

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

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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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The equation of curve is
x² – y² = a²
Differentiating both sides w.r.t. x,

2 straight x minus 2 straight y dy over dx equals 0

straight x minus straight y dy over dx equals 0

which is required differential equation.

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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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