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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.

95 Views

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

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28. Show that the function y = ae^2X + be^–x is a solution of differential equation

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

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29. In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:
$straight y space equals space straight e to the power of straight x plus 1$ : $straight y to the power of straight n minus straight y apostrophe space equals space 0$

y = e^x + 1 ;   ...(1)
∴ y' = e^x    ...(2)
and    y'' = e^x    ...(3)
L.H.S. = y'' – y'
= e^x – e^x    [∵ of (2), (3)]
= 0 = R.H.S.
∴ y = e^x + 1 is a solution of y'' – y' = 0