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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 ?

Let P be the principal at any time t. According to the given problem,

dP over dt space equals space open parentheses 5 over 100 close parentheses space cross times space straight P

therefore space space space space space space space dP over dt space equals space straight P over 20

Separating the variables, we get

dP over straight P space equals dt over 20

Integrating,

integral dP over straight P equals space integral dt over 20

therefore space space space space space space space space space space space space space space space log space straight P space equals space straight t over 20 plus straight c subscript 1 space space space or space straight P space space equals space straight e to the power of straight t over 20 plus straight c subscript 1 end exponent

straight P space equals space straight e to the power of straight t over 20 end exponent. space straight e to the power of straight c subscript 1 end exponent

straight P space equals space straight c space straight e to the power of straight t over 20 end exponent space left parenthesis where space straight e to the power of straight c subscript 1 end exponent space equals space straight c right parenthesis

...(1)
Now

straight P space equals space 1000 comma space space space when space straight t space equals space 0

therefore space space space space space space 1000 space equals space ce to the power of 0 space space space space rightwards double arrow space space space space 1000 space equals space space straight c

Putting c = 1000 in (1), we get

straight P space equals space 1000 space straight e to the power of straight t over 20 end exponent

Let t years be the time required to double the principal. Then

2000 space equals space 1000 space straight e to the power of straight t over 20 end exponent space space space space space rightwards double arrow space space space space 2 space equals space straight e to the power of straight t over 20 end exponent space space space rightwards double arrow space space straight t over 20 space equals space log subscript straight e squared space space space rightwards double arrow space space space straight t space equals space 20 space log subscript straight e 2.

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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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