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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.

The given differential equation is
y' = e^x sin x or

dy over dx space equals space straight e to the power of straight x space sin space straight x

Separating the variables, we get,

dy space equals space straight e to the power of straight x space sinx space dx

Integrating,

integral space dy space equals space integral straight e to the power of straight x space sinx space dx

therefore space space space space space space space space space space space space space space space space space space straight y space equals space fraction numerator 1 over denominator 1 plus 1 end fraction straight e to the power of straight x space left parenthesis sinx minus space cosx right parenthesis space plus space straight c

open square brackets because space space integral straight e to the power of ax space sin space straight b space straight x space dx space equals space fraction numerator 1 over denominator straight a squared plus straight b squared end fraction straight e to the power of ax space left parenthesis straight a space sin space bx space minus space straight b space cos space bx right parenthesis close square brackets

therefore space space space space space straight y space space equals 1 half straight e to the power of straight x left parenthesis sinx space minus space cosx right parenthesis space plus straight c

...(1)
Now curve passes through (0, 0)

therefore space space space 0 space equals space 1 half straight e to the power of 0 left parenthesis sin space 0 space minus cos space 0 right parenthesis space space plus straight c space space space rightwards double arrow space space space 0 space equals space 1 half cross times space left parenthesis 0 minus 1 right parenthesis space plus straight c

therefore space space space space space space space space straight e space equals space 1 half

therefore space space space from space left parenthesis 1 right parenthesis comma space space straight y space equals space 1 half straight e to the power of straight x left parenthesis sinx space minus space cosx right parenthesis space plus space 1 half comma space which space is space required space solution. space

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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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Previous Year Papers

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Long Answer Type

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

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