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Differential Equations

Short Answer Type

311.

If space straight y space equals space Pe to the power of ax plus Qe to the power of bx

show that

straight d to the power of 2 straight y end exponent over dx squared minus left parenthesis straight a plus straight b right parenthesis dy over dx plus aby space equals 0

111 Views

312.

If space straight x equals cost left parenthesis 3 minus 2 cos squared straight t right parenthesis space and space straight y equals sint left parenthesis 3 minus 2 sin squared straight t right parenthesis comma space find space the space value space of space dy over dx at space straight t equals straight pi over 4.

107 Views

313.

Find the particular solution of the differential equation $log open parentheses dy over dx close parentheses equals 3 straight x plus 4 straight y comma space given space that space straight y space equals space 0 comma space when space straight x equals space 0.$

120 Views

314.

Write the degree of the differential equation $straight x cubed open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses squared space plus space open parentheses dy over dx close parentheses to the power of 4 space equals space 0$

180 Views

315.

The amount of pollution content added in air in city due to x-diesel vehicles is given by $straight P left parenthesis straight x right parenthesis space equals space 0.005 straight x cubed plus 0.02 straight x squared plus 30 straight x.$ Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions.

225 Views

316.

Show that the function f in $straight A space equals space straight R to the power of minus open curly brackets 2 over 3 close curly brackets space defined space as space straight f left parenthesis straight x right parenthesis space equals space fraction numerator 4 straight x plus 3 over denominator 6 straight x minus 4 end fraction space is space one minus one space and space onto. space Hence space find space straight f to the power of negative 1 end exponent.$

130 Views

317.
Differentiate the following function with respect to x:
$left parenthesis log space straight x right parenthesis to the power of straight x plus straight x to the power of log straight x$

$Let space straight y space equals left parenthesis logx right parenthesis to the power of straight x plus straight x to the power of logx space space space space space space space space... left parenthesis 1 right parenthesis Now space let space straight y subscript 1 equals left parenthesis log space straight x right parenthesis to the power of straight x space and space straight y subscript 2 equals straight x to the power of logx rightwards double arrow space space straight y space equals space straight y subscript 1 plus straight y subscript 2 space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis$
Differentiating (2), w.r.t.x,
$dy over dx equals dy subscript 1 over dx plus dy subscript 2 over dx space space space space space space space space space space... left parenthesis 3 right parenthesis$

Now consider

straight y subscript 1 space equals space left parenthesis log space straight x right parenthesis to the power of straight x

Taking log on both sides,

logy subscript 1 space equals space straight x space log left parenthesis log space straight x right parenthesis

Differentiating, w.r.t.x, we get

1 over straight y subscript 1 dy subscript 1 over dx equals straight x cross times 1 over logx cross times 1 over straight x plus 1 cross times log left parenthesis log space straight x right parenthesis rightwards double arrow space space dy subscript 1 over dx space equals space straight y subscript 1 open parentheses 1 over logx plus log left parenthesis log space straight x right parenthesis close parentheses rightwards double arrow dy subscript 1 over dx equals left parenthesis logx right parenthesis to the power of straight x open parentheses fraction numerator 1 over denominator log space straight x end fraction plus log space left parenthesis log space straight x right parenthesis close parentheses space space space space space space... left parenthesis 4 right parenthesis

Now comma space consider space straight y subscript 2 space equals straight x to the power of logx logy subscript 2 space equals space left parenthesis log space straight x right parenthesis left parenthesis log space straight x right parenthesis space equals left parenthesis log space straight x right parenthesis squared Differentiating space straight w. straight r. straight t. straight x comma space we space get 1 over straight y subscript 2 dy subscript 2 over dx equals 2 left parenthesis log space straight x right parenthesis cross times 1 over straight x rightwards double arrow space dy subscript 2 over dx space equals straight y subscript 2 open parentheses fraction numerator 2 logx over denominator straight x end fraction close parentheses equals straight x to the power of logx open parentheses fraction numerator 2 logx over denominator straight x end fraction close parentheses... left parenthesis 5 right parenthesis

Using equations (3), (4) and (5), we get:

dy over dx equals left parenthesis log space straight x right parenthesis to the power of straight x open square brackets 1 over logx plus log left parenthesis log space straight x right parenthesis close square brackets plus straight x to the power of logx open parentheses fraction numerator 2 logx over denominator straight x end fraction close parentheses

142 Views

318.

If space straight y space equals space log space open square brackets straight x space plus space square root of straight x squared plus space straight a squared end root close square brackets comma space show space that space left parenthesis straight x squared space plus space straight a squared right parenthesis space fraction numerator straight d squared straight y over denominator dx squared end fraction space plus straight x dy over dx space equals space 0

186 Views

319.

If $straight x equals straight a space sint space and space straight y space equals space straight a open parentheses cost plus log space tan straight t over 2 close parentheses space find space fraction numerator straight d squared straight y over denominator dx squared end fraction$

500 Views

Long Answer Type

320.

Show that the differential equation $2 ye to the power of straight x divided by straight y end exponent dx plus left parenthesis straight y minus 2 straight x space straight e to the power of straight x divided by straight y end exponent right parenthesis space dy space equals space 0$ is homogeneous. Find the particular solution of this differential equation, given that x = 0 when y = 1.