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Differential Equations

Long Answer Type

161.

For the differential equation $xy dy over dx space equals space left parenthesis straight x plus 2 right parenthesis thin space left parenthesis straight y plus 2 right parenthesis comma$ find the solution curve passing through the point (1, -1).

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162. Find the equation of a curve passing through the point (0, – 2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

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163. At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, – 3). Find the equation of the curve given that it passes through ( 2, 1).

Let y = f(x) be equation of curve.
Now

dy over dx

is the slope of the tangent to the curve at the point (x, y)
From the given condition,

dy over dx space equals space 2 open parentheses fraction numerator negative 3 minus straight y over denominator negative 4 minus straight x end fraction close parentheses space space or space space space dy over dx space equals space 2 open parentheses fraction numerator straight y plus 3 over denominator straight x plus 4 end fraction close parentheses

Separating the variables, we get,

fraction numerator 1 over denominator straight y plus 3 end fraction dy space equals space fraction numerator 2 over denominator straight x plus 4 end fraction dx

Integrating,

integral fraction numerator 1 over denominator straight y plus 3 end fraction space dy space equals space 2 space integral fraction numerator 1 over denominator straight x plus 4 end fraction dx

therefore space space space log space open vertical bar straight y plus 3 close vertical bar space equals space 2 space log space open vertical bar straight x plus 4 close vertical bar plus straight c

...(1)
Since curve passe through (-2, 1)

therefore space space space log space open vertical bar 1 plus 3 close vertical bar space equals space 2 space log space open vertical bar negative 2 plus 4 close vertical bar space plus space straight c therefore space space log space 4 space equals space 2 space log space 2 space plus straight c space space space rightwards double arrow space space space space 2 space log space 2 space space equals space 2 space log space 2 plus space straight c space space rightwards double arrow space space straight c space equals space 0 therefore space space from space left parenthesis 1 right parenthesis comma space space log space open vertical bar straight y plus 3 close vertical bar space equals space 2 space log space open vertical bar straight x plus 4 close vertical bar or space space log space open vertical bar straight y plus 3 close vertical bar space equals space log space open vertical bar straight x plus 4 close vertical bar squared therefore space space space space open vertical bar straight y plus 3 close vertical bar space equals space open vertical bar straight x plus 4 close vertical bar squared space space space or space space space straight y plus 3 space equals space left parenthesis straight x plus 4 right parenthesis squared

which is required equation of curve.

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164. Find the equation of the curve passing through the point

open parentheses 0 comma space straight pi over 4 close parentheses

whose differential equation is sin x cos y dx + cos x sin y dy = 0.

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165. Find the equation of a curve passing through the point (0, 0) and whose differential equation is y' = e^x sin x.

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166.

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.

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167. In a bank, principal increases continuously at the rate of 5% per year. In how many years Rs. 1000 double itself ?

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168. In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs. 100 double itself in 10 years.

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169. In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs. 100 double itself in 10 years. (e^0·5 = 1·648).

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170.

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

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