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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).

The given differential equation is

straight x space straight y dy over dx space equals space left parenthesis straight x plus 2 right parenthesis thin space left parenthesis straight y plus 2 right parenthesis

Separating the variables, we get,

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

Integrating,

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

therefore space space space space integral fraction numerator left parenthesis straight y plus 2 right parenthesis space minus space 2 over denominator straight y plus 2 end fraction dy space equals space integral open parentheses straight x over straight x plus 2 over straight x close parentheses dx

therefore space space space space integral open parentheses 1 minus fraction numerator 2 over denominator straight y plus 2 end fraction close parentheses space dy space equals space integral open parentheses 1 plus 2 over straight x close parentheses dx

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

...(1)
Since the curve passes through (1, -1)

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

straight y minus straight x plus 2 space equals space 2 space log space open vertical bar straight x space left parenthesis straight y plus 2 right parenthesis close vertical bar

straight y minus straight x plus 2 space equals space log space open vertical bar straight x space left parenthesis straight y plus 2 right parenthesis close vertical bar squared

straight y minus straight x plus 2 space equals space log space open square brackets straight x squared space left parenthesis straight y plus 2 right parenthesis squared close square brackets

is the required solution.

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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.

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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

Test Series

Test Yourself

Long Answer Type

161. For the differential equation find the solution curve passing through the point (1, -1).

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).