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Differential Equations

Short Answer Type

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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Multiple Choice Questions

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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Short Answer Type

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.

The given equation is $straight y space equals space ke to the power of sin to the power of negative 1 end exponent straight x end exponent plus 3$ ...(1)
$therefore space space space dy over dx equals space straight k space straight e to the power of sin to the power of negative 1 end exponent straight x end exponent. fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction space or space space dy over dx space equals space left parenthesis straight y minus 3 right parenthesis. space fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets$
$or space space square root of 1 minus straight x squared end root space dy over dx equals straight y minus 3 comma space$ which is required differential equation.