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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.

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

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 Let space straight f left parenthesis straight x subscript 1 right parenthesis space equals space straight f left parenthesis straight x subscript 2 right parenthesis rightwards double arrow fraction numerator 4 straight x subscript 1 plus 3 over denominator 6 straight x subscript 1 minus 4 end fraction equals fraction numerator 4 straight x subscript 2 plus 3 over denominator 6 straight x subscript 2 minus 4 end fraction rightwards double arrow space space 24 straight x subscript 1 straight x subscript 2 minus 16 straight x subscript 1 plus 18 straight x subscript 2 minus 12 space equals space 24 straight x subscript 1 straight x subscript 2 plus 18 straight x subscript 1 minus 16 straight x subscript 2 minus 12 rightwards double arrow space space 18 straight x subscript 2 plus 16 straight x subscript 2 space equals space 18 straight x subscript 1 plus 16 straight x subscript 1 rightwards double arrow 34 straight x subscript 2 space equals 34 straight x subscript 1 rightwards double arrow straight x subscript 1 space equals space straight x subscript 2

Since,

fraction numerator 4 straight x plus 3 over denominator 6 straight x minus 4 end fraction

is a real number, therefore, for every y in the co-domain of f, there exists a number x in

straight R to the power of minus open curly brackets 2 over 3 close curly brackets space such space that space straight f left parenthesis straight x right parenthesis space equals space straight y space equals space fraction numerator 4 straight x plus 3 over denominator 6 straight x minus 4 end fraction

Therefore, f(x) is onto.
Hence,

straight f to the power of negative 1 end exponent

exists.
Now,

let space straight y space equals space fraction numerator 4 straight x plus 3 over denominator 6 straight x minus 4 end fraction rightwards double arrow 6 xy minus 4 straight y space equals space 4 straight x plus 3 rightwards double arrow space 6 xy minus 4 straight x space equals space 4 straight y plus 3 rightwards double arrow straight x left parenthesis 6 straight y minus 4 right parenthesis space equals space 4 straight y plus 3 rightwards double arrow space space space straight x space equals space fraction numerator 4 straight y plus 3 over denominator 6 straight y minus 4 end fraction rightwards double arrow space space straight y equals space fraction numerator 4 straight x plus 3 over denominator 6 straight x minus 4 end fraction space space left square bracket interchanging space the space variables space straight x space and space straight y right square bracket rightwards double arrow straight f to the power of negative 1 end exponent 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 left square bracket put space straight y space equals space straight f to the power of negative 1 end exponent left parenthesis straight x right parenthesis right square bracket

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

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

316. Show that the function f in

Long Answer Type