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

straight x equals cost space left parenthesis 3 minus 2 cos squared straight t right parenthesis and straight y equals sint left parenthesis 3 minus sin squared straight t right parenthesis We space need space to space find space dy over dx colon dy over dx equals fraction numerator begin display style dy over dx end style over denominator begin display style dx over dt end style end fraction Let space us space find space dx over dt colon straight x equals cost left parenthesis 3 minus 2 cos squared straight t right parenthesis dx over dt equals cost left parenthesis 4 cost space sint right parenthesis plus left parenthesis 3 minus 2 cos squared straight t right parenthesis left parenthesis negative sint right parenthesis rightwards double arrow dx over dt equals negative 3 sint plus 4 cos squared tsint plus 2 cos squared tsint Let space us space find space dy over dt colon straight y equals sint left parenthesis 3 minus 2 sin squared straight t right parenthesis dy over dt equals sint left parenthesis negative 4 sint space cost right parenthesis plus left parenthesis 3 minus 2 sin squared straight t right parenthesis space left parenthesis cost right parenthesis rightwards double arrow dy over dt equals 3 cost minus 4 sin squared tcost minus 2 sin squared tcost Thus comma space dy over dx equals fraction numerator 3 cost minus 4 sin squared tcost minus 2 sin squared tcost over denominator negative 3 sint plus 4 cos squared tsint plus 2 cos squared tsint end fraction rightwards double arrow space dy over dx equals fraction numerator 3 cost minus 6 sin squared tcost over denominator negative 3 sint plus 6 cos squared tsint end fraction rightwards double arrow dy over dx equals fraction numerator 3 cost left parenthesis 1 minus 2 sin squared straight t right parenthesis over denominator negative 3 sint left parenthesis 1 minus 2 cos squared straight t right parenthesis end fraction rightwards double arrow dy over dx equals fraction numerator 3 cost left parenthesis 1 minus 2 sin squared straight t right parenthesis over denominator 3 sint left parenthesis 2 cos squared straight t minus 1 right parenthesis end fraction rightwards double arrow dy over dx equals fraction numerator cost space over denominator sint end fraction open square brackets because 2 cos squared straight t minus 1 equals 1 minus 2 sin squared straight t close square brackets rightwards double arrow dy over dx equals cot space straight t rightwards double arrow open parentheses dy over dx close parentheses subscript straight t equals straight pi over 4 end subscript equals cot straight pi over 4 equals 1

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:
$left parenthesis log space straight x right parenthesis to the power of straight x plus straight x to the power of log straight x$

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.

135 Views