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# Differential equations: final exam

This is for Differential equations: course

2011:

1. Verify that the function $y=cx^{2}$ is a solution of the differential equation: $xy^{\prime}=2y.$ Are there any others?
2. Find all curves perpendicular to the family of curves: $xy^{2}=C.$
3. Suppose point $T$ goes along the line $x=1$ while dragging point $P$ on the $xy$-plane by a string $PT$ of length 1. Suppose $T$ starts at $(1,0)$ and $P$ at $(2,0)$. Find the path of $P.$
4. Solve the following differential equation: $y^{\prime\prime}+2y^{\prime }+4y=0.$
5. Provide the power series solution for $y^{\prime}+y=1.$
6. Use Euler's method with 4 steps to estimate the solution of the initial values problem: $y^{\prime}=2x-y,y(0)=1,$ on the interval $[0,1].$
7. Set up (don't solve) initial values problems for the following two situations: (a) An object is thrown up from a building of height $h$ at 45 degrees with speed $s;$ (b) An object thrown travels for 2 seconds and then hits the ground at 45 degrees and speed $s.$
8. Set up and solve the differential equation that describes the motion of an object of mass $M$ placed on top of a spring with Hooke constant $k$ standing vertically on the ground$.$
9. (a) Describe the predator-prey model. (b) Set up a system of differential equations for the model and find its equilibria.