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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.

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

dy over dx

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

dy over dx cross times straight y space equals space straight x space space space space space space space space space space space space space space space space space space space space or space space space space space dy over dx space equals space straight x over straight y

From the given condition,

dy over dx cross times straight y space equals straight x space space space space space space or space space space dy over dx space equals space straight x over straight y

Separating the variables and integrating,

integral space straight y space dy space equals space integral straight x space dx space space space space space space space space space space space or space space space space space straight y squared over 2 space equals space straight x squared over 2 plus straight c

...(1)
Since it passes through (0, -2)

therefore space space space space space fraction numerator left parenthesis negative 2 right parenthesis squared over denominator 2 end fraction space equals space fraction numerator left parenthesis 0 right parenthesis squared over denominator 2 end fraction plus straight c space space space space rightwards double arrow space space space space 2 space equals space straight c

therefore space space space from space left parenthesis 1 right parenthesis comma space space space space straight y squared over 2 space equals space straight x squared over 2 plus 2 space space space or space space space space straight y squared space equals space straight x squared plus 4

which is required equation of curve.

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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.

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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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