Ordinary differential equations UE Past Papers Questions.


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(1916) Question Category: Mathematical Calculation

Draw a direction field for the given differential equation. Based on the direction field, determine the behaviour of y at t ightarrow infty

  1. y`=3-2y
  2. y`=3+2y
  3. y`=1+2y

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(1917) Question Category: Mathematical Calculation

Consider the initial value problem y`+frac{1}{2}y=2cos t, y(0)=-1. Find the coordinates of the first local maximum point of the solution for t>0.

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(1918) Question Category: Mathematical Calculation

Solve the initial value problem y"-y`-2y=0, y(0)=alpha . Then find alpha so that the solution approaches zero as t ightarrow infty.

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(1919) Question Category: Mathematical Calculation

Consider the initial value problem 4y"+12y`+9y=0, y(0)=1, y`(0)=-4.

  1. Solve the initial value problem
  2. Determine where the solution has the value zero
  3. Determine the coordinates (t_{0},y_{0}) of the minimum point.

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(1920) Question Category: Mathematical Calculation

  1. Solve the Gompertz equation frac{dy}{dt}=ry, lnleft ( frac{K}{y} ight ), y(0)=y_{0} Hint : You may wish to let u=ln(frac{y}{K}).
  2. Suppose that r=0.71 per year, K=80.5 imes 10^{6}kg and frac{y_0}{K}=0.25, use the Gompertz model in (a) to find the predicted value of y(2).
  3. For the same data as in part(b), use the Gompertz model to find the time r at which y(r)=0.75K

Answer / Solution

UNSOLVED

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