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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

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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.

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

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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.

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316.

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

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317.

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

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

Given,

$straight y space equals space log space open square brackets straight x space plus space square root of straight x squared space plus space straight a squared end root close square brackets space straight y space equals space log space open square brackets straight x space plus space square root of straight x squared space plus straight a squared end root close square brackets...... space left parenthesis straight i right parenthesis Differentiating space left parenthesis 1 right parenthesis space straight w. straight r. straight t space space straight x comma space we space get dy over dx space equals space fraction numerator 1 plus fraction numerator straight x over denominator square root of straight x squared plus straight a squared end root end fraction over denominator straight x space plus space square root of straight x squared space plus space straight a squared end root end fraction rightwards double arrow space dy over dx space equals space fraction numerator 1 over denominator square root of straight x squared space plus space straight a squared end root end fraction space space space space space..... space left parenthesis ii right parenthesis rightwards double arrow space straight x dy over dx space equals space fraction numerator straight x over denominator square root of straight x squared space plus space straight a squared end root end fraction space space space space space space.... space left parenthesis iii right parenthesis Again comma space differentiating space left parenthesis ii right parenthesis space straight w. straight r. straight t space straight x comma space we space get fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space fraction numerator negative begin display style fraction numerator 2 straight x over denominator 2 left parenthesis straight x squared space plus space straight a squared right parenthesis to the power of begin display style 1 half end style end exponent end fraction end style over denominator left parenthesis straight x squared space plus space straight a squared right parenthesis end fraction rightwards double arrow space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space minus fraction numerator straight x over denominator left parenthesis straight x squared space plus space straight a squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction rightwards double arrow space left parenthesis straight x squared space plus space straight a squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space minus fraction numerator straight x over denominator square root of straight x squared space plus space straight a squared end root end fraction space space... space left parenthesis iv right parenthesis adding space equation space left parenthesis iii right parenthesis space and space left parenthesis iv right parenthesis comma space we space get 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 space straight x dy over dx space equals space minus fraction numerator straight x over denominator square root of straight x squared plus straight a squared end root end fraction space space plus fraction numerator straight x over denominator square root of straight x squared space plus space straight a squared end root end fraction space equals space 0 rightwards double arrow space left parenthesis space 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 space 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.

135 Views