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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

y e^x + x² = C

The given differential equation is

straight e to the power of straight x dy space plus space left parenthesis straight y space straight e to the power of straight x space plus space 2 straight x right parenthesis space dx space equals space 0

straight e to the power of straight x dy over dx plus space straight y space straight e to the power of straight x space plus space 2 space straight x space equals 0 space space space space or space space space dy over dx plus straight y plus fraction numerator 2 straight x over denominator straight e to the power of straight x end fraction space equals space 0 space space or space space dy over dx plus straight y space equals space minus 2 xe to the power of negative straight x end exponent

Comparing it with

dy over dx plus Py space equals space straight Q comma

we get, P = 1,

straight Q space equals negative 2 xe to the power of negative straight x end exponent

therefore space space space space space space space space space space integral straight P space dx space equals space integral space 1 space dx space equals space apostrophe straight x comma space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of straight x

Solution of given differential equation is

ye space to the power of integral straight P space dx end exponent space equals space integral straight Q space straight e to the power of integral straight P space dx end exponent dx plus straight C space space space or space space space straight y space straight e to the power of straight x space equals space integral left parenthesis negative 2 straight x space straight e to the power of straight x right parenthesis. space straight e to the power of straight x dx plus straight C

straight y space straight e to the power of straight x space equals space minus 2 space integral space straight x space dx space plus space straight C space space space space or space space space straight y space straight e to the power of straight x space equals space minus straight x squared plus straight C

space ye to the power of straight x plus straight x squared space equals space straight C

therefore space space left parenthesis straight C right parenthesis space is space correct space answer.

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Short Answer Type

302.

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

1130 Views

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.

761 Views

304.

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

1357 Views

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

516 Views

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.

513 Views

307.

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

783 Views

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.

438 Views

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