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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.

2 ye to the power of straight x over straight y end exponent dx space plus space open parentheses straight y minus 2 xe to the power of straight x over straight y end exponent close parentheses dy space space equals space 0 rightwards double arrow space dx over dy space equals space fraction numerator 2 xe to the power of begin display style straight x over straight y end style end exponent minus straight y over denominator 2 ye to the power of begin display style straight x over straight y end style end exponent end fraction Given space differential space equation space is space straight a space homogenous space differential space equation. therefore comma put space space straight x space equals space vy dx over dy space equals space straight v space plus space straight y dv over dy straight v space plus space straight y dv over dy space space space equals space fraction numerator 2 ve to the power of straight v minus 1 over denominator 2 straight e to the power of straight v end fraction rightwards double arrow space space straight y dv over dy space equals space fraction numerator 2 ve to the power of straight v minus 1 over denominator 2 straight e to the power of straight v end fraction space minus space straight v rightwards double arrow space space straight y dv over dy space equals space minus fraction numerator 1 over denominator 2 straight e to the power of straight v end fraction rightwards double arrow space space 2 straight e to the power of straight v dv space equals negative 1 over straight y dy Integrating space on space both space sides rightwards double arrow space space 2 integral straight e to the power of straight v equals negative integral 1 over straight y dy rightwards double arrow space 2 straight e to the power of straight v space space equals space log space open vertical bar straight y close vertical bar space plus space log space straight C rightwards double arrow space 2 straight e to the power of straight v space equals space log space space open vertical bar straight c over straight y close vertical bar rightwards double arrow space 2 straight e to the power of fraction numerator straight x over denominator straight y space end fraction end exponent equals space log space open vertical bar straight c over straight y close vertical bar Given space that space at space straight x equals 0 comma space straight y equals 1 2 straight e to the power of 0 space space equals space log open vertical bar straight c over 1 close vertical bar rightwards double arrow space straight C space equals space straight e squared therefore comma 2 straight e to the power of straight x over straight y end exponent space equals space log space straight e squared over straight y rightwards double arrow space space log space straight y equals negative 2 straight e to the power of straight x over straight y end exponent plus space 2 rightwards double arrow space straight y space equals space straight e to the power of 2 minus 2 straight e to the power of straight x over straight y end exponent end exponent

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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.

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

315 Views

Short Answer Type

310.

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