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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 $log open parentheses dy over dx close parentheses equals 3 straight x plus 4 straight y comma space given space that space straight y space equals space 0 comma space when space straight x equals space 0.$

Consider the differential equation,
$log open parentheses dy over dx close parentheses equals 3 straight x plus 4 straight y$
Taking exponent on both the sides, we have
$straight e to the power of log open parentheses dy over dx close parentheses end exponent space equals space straight e to the power of 3 straight x plus 4 straight y end exponent space rightwards double arrow dy over dx equals straight e to the power of 3 straight x plus 4 straight y end exponent space rightwards double arrow dy over dx equals straight e to the power of 3 straight x end exponent. straight e to the power of 4 straight y end exponent rightwards double arrow dy over straight e to the power of 4 straight y end exponent equals straight e to the power of 3 straight x end exponent. dx$
Integration in both the sides, we have
$integral dy over straight e to the power of 4 straight y end exponent space equals space integral straight e to the power of 3 straight x end exponent dx fraction numerator straight e to the power of negative 4 straight y end exponent over denominator negative 4 end fraction equals straight e to the power of 3 straight x end exponent over 3 plus straight C We space need space to space find space the space particular space solution We space have comma space straight y space equals space 0 comma space when space straight x space equals space 0 fraction numerator 1 over denominator negative 4 end fraction space equals 1 third plus straight C rightwards double arrow straight C space equals negative 1 fourth minus 1 third rightwards double arrow straight C space equals space fraction numerator negative 3 minus 4 over denominator 12 end fraction equals negative 7 over 12 Thus comma space the space solution space is space straight e to the power of 3 straight x end exponent over 3 plus straight e to the power of negative 4 straight y end exponent over 4 equals 7 over 12$

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.