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Differential Equations

Multiple Choice Questions

301. The general solution of the differential equation e^x dy + (ye^x + 2x) dx = 0 is

x e^y + x² = C
x e^y + y² = C
y e^x + x² = C
y e^x + x² = C

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Short Answer Type

302.

Find the particular solution of the differential equation
$dy over dx equals negative fraction numerator x plus y space c o s space x over denominator 1 plus s i n space x space end fraction space g i v e n space t h a t space y equals 1 space w h e n space x equals 0$

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303.

Find the particular solution of the differential equation

2ye^x/y dx+ (Y-2xe^x/y) dy =0

Given that x=0 when y=1.

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304.

Find the differential equation of the family of lines passing through the origin.

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305.

If space straight y space equals space straight e to the power of ax. cosbx comma space then space prove space that space space space fraction numerator straight d squared straight y over denominator dx squared end fraction minus 2 straight a dy over dx plus left parenthesis straight a squared plus straight b squared right parenthesis straight y space equals space 0

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306. $If space straight x to the power of straight x plus straight x to the power of straight y plus straight y to the power of straight x space equals space straight a to the power of straight b comma space space then space find space dy over dx.$

straight x to the power of straight x plus straight x to the power of straight y plus straight y to the power of straight x space equals straight a to the power of straight b space space space... left parenthesis straight i right parenthesis Let space straight u space equals space straight x to the power of straight x log space straight u space equals space straight x space log space straight x 1 over straight u. du over dx equals straight x.1 over straight x plus log space straight x therefore space space du over dx equals straight x to the power of straight x left parenthesis 1 plus log space straight x right parenthesis Let space straight v equals space straight x to the power of straight y log space straight v space equals space straight y space log space straight x 1 over straight v. dv over dx space equals space open parentheses straight y over straight x plus log space straight x dy over dx close parentheses therefore space dv over dx equals straight x to the power of straight y open parentheses straight y over straight x plus logx dy over dx close parentheses Let space straight w space equals straight y to the power of straight x Log space straight w space equals space straight x space log space straight y 1 over straight w. dw over dx equals space open parentheses straight x over straight y. dy over dx plus log space straight y close parentheses therefore space dw over dx equals straight y to the power of straight x open parentheses log space straight y space plus space straight x over straight y. dy over dx close parentheses

left parenthesis straight i right parenthesis space can space be space writeen space as straight u plus straight v plus straight w space equals space straight a to the power of straight b du over dx plus dv over dx plus dw over dx equals 0 rightwards double arrow space space straight x to the power of straight x left parenthesis 1 plus logx right parenthesis plus straight x to the power of straight y open parentheses straight y over straight x plus logx dy over dx close parentheses plus straight y to the power of straight x open parentheses logy plus straight x over straight y dy over dx close parentheses space equals space 0 rightwards double arrow space space straight x to the power of straight x plus straight x to the power of straight x logx plus straight x to the power of straight y. straight y over straight x plus straight x to the power of straight y. logx. dy over dx plus straight y to the power of straight x. logy plus straight y to the power of straight x. straight x over straight y. dy over dx equals 0 rightwards double arrow space space dy over dx open parentheses straight x to the power of straight y. logx space plus space straight y to the power of straight x. straight x over straight y close parentheses space equals space straight x to the power of straight x plus straight x to the power of straight x logx plus straight x to the power of straight y. straight y over straight x plus straight y to the power of straight x. log space straight y rightwards double arrow dy over dx open parentheses straight x to the power of straight y. logx space plus space xy to the power of straight x minus 1 end exponent close parentheses space equals space open parentheses straight x to the power of straight x plus straight x to the power of straight x logx plus yx to the power of straight y minus 1 end exponent plus straight y to the power of straight x. log space straight y close parentheses therefore space space dy over dx equals fraction numerator left parenthesis straight x to the power of straight x plus straight x to the power of straight x logx space plus yx to the power of straight y minus 1 end exponent plus straight y to the power of straight x. log space straight y right parenthesis over denominator left parenthesis straight x to the power of straight y. logx plus xy to the power of straight x minus 1 end exponent right parenthesis end fraction

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307.

If $straight x equals straight a space sin space 2 straight t left parenthesis 1 plus cos space 2 straight t right parenthesis space and space straight y space equals straight b space cos space 2 straight t left parenthesis 1 minus cos space 2 straight t right parenthesis comma space then space find dy over dx space at space straight t space equals straight pi over 4.$

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Long Answer Type

308.

Find the particulars solution of the differential equation $straight x squared dy space equals space left parenthesis 2 xy plus straight y squared right parenthesis dx comma space$ given that y =1 when x = 1.

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309.

Find the particulars solution of the differential equation $left parenthesis 1 plus straight x squared right parenthesis dy over dx equals open parentheses straight e to the power of straight m space tan to the power of negative 1 straight x end exponent end exponent minus straight y close parentheses comma space given space that space straight y space equals space 1 space when space straight x space equals space 0.$

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Short Answer Type

310.

Solve the differential equation $left parenthesis 1 plus straight x squared right parenthesis dy over dx plus straight y equals straight e to the power of tan to the power of negative 1 end exponent straight x end exponent$

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