Ordinary differential equations questions.

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#	Question
1	Draw a direction field for the given differential equation. Based on the direction field, determine the behaviour of y at $t ightarrow infty$ $y`=3-2y$ $y`=3+2y$ $y`=1+2y$ Mathematical Calculation
2	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. Mathematical Calculation
3	Solve the initial value problem $y"-y`-2y=0, y(0)=alpha$ . Then find $alpha$ so that the solution approaches zero as $t ightarrow infty$ . Mathematical Calculation
4	Consider the initial value problem $4y"+12y`+9y=0, y(0)=1, y`(0)=-4$ . Solve the initial value problem Determine where the solution has the value zero Determine the coordinates $(t_{0},y_{0})$ of the minimum point. Mathematical Calculation
5	Solve the Gompertz equation $frac{dy}{dt}=ry, lnleft ( frac{K}{y} ight ), y(0)=y_{0}$ Hint : You may wish to let $u=ln(frac{y}{K})$ . 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 $y(2)$ . For the same data as in part(b), use the Gompertz model to find the time r at which $y(r)=0.75K$ Mathematical Calculation
6	Solve the equation $x^{5}y"+6x^{5}y`+9x^{5}y=e^{-3x}$ by variation of parameters. Mathematical Calculation
7	Find a series solution of equation $y{}'+y=0, -infty <x<infty$ Mathematical Calculation
8	Use the Laplace transform to solve the initial value problem $frac{dy}{dx}+3y=13sin, 2t; y(0)=6$ Mathematical Calculation
9	Consider the system $x{}`=egin{pmatrix} 1 &1 \ 4& 1 end{pmatrix}x$ Find its general solution and plot several trajectories in the phase plane. Mathematical Calculation
10	Solve by variation of parameters $x^{2}y{}'-2xy` +2y=x^{3}lnx$ Mathematical Calculation