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Differential Equations

Short Answer Type

21.

Determine the order and degree of the differential equation:
$square root of straight x fraction numerator straight d to the power of 4 straight y over denominator dx to the power of 4 end fraction plus 5 open parentheses dy over dx close parentheses squared space equals 2 straight y squared minus straight x$

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Multiple Choice Questions

22. The degree of the differential equation

open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses cubed plus open parentheses dy over dx close parentheses squared plus sin open parentheses dy over dx close parentheses plus 1 space space equals space 0

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23. The degree of the differential equation

2 straight x squared fraction numerator straight d squared straight y over denominator dx squared end fraction minus 3 dy over dx plus straight y space equals space 0

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Short Answer Type

24. Verify that y = A cor x – B sin x is a solution of the differential equation

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

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25. Verify that the function y = a cos x + b sin x, where a, b ∊ R is a solution of the differential equation

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

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Long Answer Type

26. Verify that y - 3 cos (log x) + 4 sin (log x) is a solution of the differential equation

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

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Short Answer Type

27. Verify that the function y = e^–3x is a solution of the differential equation $fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx minus 6 straight y space equals space 0$

$straight y space equals space straight e to the power of negative 3 straight x end exponent$ ...(1)
$therefore$ $dy over dx space equals space straight e to the power of negative 3 straight x end exponent. space left parenthesis negative 3 right parenthesis space space space space space space space or space space space space space dy over dx equals negative 3 straight e to the power of negative 3 straight x end exponent$ ...(2)

$fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative 3 space straight e to the power of negative 3 straight x end exponent. space left parenthesis negative 3 right parenthesis space space space or space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space 9 straight e to the power of negative 3 straight x end exponent$ ...(3)
L.H.S. = $fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx minus 6 straight y space equals space 9 straight e to the power of negative 3 straight x end exponent minus 3 straight e to the power of 3 straight x end exponent minus 6 straight e to the power of negative 3 straight x end exponent space space space space space open square brackets because space of space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis close square brackets$
= 0 - R.H.S.
$therefore space space space straight y space equals straight e to the power of negative 3 straight x end exponent$ is a solution of the differential equation
$fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx minus 6 straight y space equals space 0$