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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).

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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.

Let P denote the principal at any time t and r% per annum be the interest rate..
From the given condition,

dP over dt space equals space left parenthesis 0.0 space straight r right parenthesis thin space straight P

Separating the variables and integrating, we get,

integral 1 over straight P dP space equals space integral 0.0 space straight r space dt

therefore space space space space space space log space straight P space equals 0.0 space straight r space straight t space plus space log space straight c space space or space space space log space straight P over straight c space equals space 0.0 space straight r space straight t

therefore

straight P space equals space straight c space straight e to the power of 0.00 straight r space straight t end exponent

...(1)
Now

straight P space equals space 100

when t = 0

therefore

100 space equals space straight c space straight e to the power of 0

rightwards double arrow space space space straight c space equals space 100

therefore space space space from space left parenthesis 1 right parenthesis comma space space space straight P space equals space 100 straight e to the power of 0.0 straight r space straight t end exponent

Now, Rs. 100 become Rs.200 in 10 years.

therefore space space space space space space space space space space space 200 space equals space 100 space straight e to the power of left parenthesis 0.0 space straight r right parenthesis space space 10 end exponent space space space space space space space space rightwards double arrow space space space space 2 space equals space straight e to the power of straight r over 10 end exponent therefore space space space space space space log subscript straight e 2 space equals space straight r over 10 space space space space space rightwards double arrow space space space space space space 0.6931 space equals space straight r over 10 space space rightwards double arrow space space space straight r space equals space 6.931 therefore space space space space space required space rate space of space interest space equals space 6.9 percent sign

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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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