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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.

The given differential equation is
sin x cos y dx + cos x sin y dy = 0
or sin x cos y dx = - cos x sin y dy
$therefore space space space space space space space space space fraction numerator negative sin space straight y over denominator cos space straight y end fraction dy space equals space fraction numerator sin space straight x over denominator cos space straight x end fraction dx$
Integrating $integral fraction numerator negative sin space straight y over denominator cos space straight y end fraction dy space equals space fraction numerator sin space straight x over denominator cos space straight x end fraction dx$
$therefore space space log space open vertical bar cos space straight y close vertical bar space plus space log space open vertical bar cos space straight x close vertical bar space equals space log space straight A space space space space space space space rightwards double arrow space space space space space log space open vertical bar cosx space cosy close vertical bar space equals space log space straight A therefore space space space open vertical bar cosx space cosy close vertical bar space equals space straight A space space space space space space space space space space space space space space space rightwards double arrow space space space space cos space straight x space cos space straight y space equals space straight c space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis$
Since the curve passes through $open parentheses 0 comma space straight pi over 4 close parentheses$
$therefore space space space cos space 0 space cos space straight pi over 4 space equals straight c space space space space space space rightwards double arrow space space space 1 space cross times space fraction numerator 1 over denominator square root of 2 end fraction space equals space straight c space space space space space rightwards double arrow space space space straight c space equals space fraction numerator 1 over denominator square root of 2 end fraction$
$therefore space space from space left parenthesis 1 right parenthesis comma space cosx space cos space straight y space space equals space fraction numerator 1 over denominator square root of 2 end fraction comma space space which space is space required space equation.$

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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

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Long Answer Type

164. Find the equation of the curve passing through the point whose differential equation is sin x cos y dx + cos x sin y dy = 0.

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.