You should write out major steps. The total score is $100.$

$(1)(12$ pts$)$ Solve the following initial value problem:

$($dy$)/($dt$),=(2$t$+3)/($y$^(2))$

y$(1),=3$

$(2)(12$ pts$)$ Use Euler's method with $\backslash$Delta t$=0\mathrm{.}5$ to approximate y$(3)$ of the initial value problem:

$($dy$)/($dt$),=$y$-$t

y$(1),=2$

$(3)(12$ pts$)$ Use extended linearity principle to find the general solution to the following linear

differential equation:

$($dy$)/($dt$)=6$y$+$sin$(3$t$)$

$(4)(12$ pts$)$ Use integrating factor method to find the general solution to the following linear

differential equation:

$($dy$)/($dt$)=(2)/($t$)$y$+$t$^(4)$

$(5)(12$ pts$)$ Solve the following partially decoupled first$-$order system:

$($dx$)/($dt$)=$x$^(2)$y

$($dy$)/($dt$)=2$y

$(6)(12$ pts$)$ Find all the equilibrium solutions of the following first$-$order system:

$($dx$)/($dt$)=$x$^(2)-$y$^(2)$

$($dy$)/($dt$)=2$x$+$y$-3$

$(7)(14$ pts$)$ Determine the bifurcation value $\backslash$mu $\_(0)$ for the following equation with parameter $\backslash$mu $.$ Then

draw the phase lines of the differential equation for $\backslash$mu $\_(0)-1,\backslash$mu $\_(0)$ and $\backslash$mu $\_(0)+1.$

$($dy$)/($dt$)=$y$^(2)-2\backslash$mu y

$(8)(14$ pts$)$ Find the solution of the following linear system with the given initial condition:

$($dx$)/($dt$)=3$x$+2$y

$($dy$)/($dt$)=3$x$+4$y

and

x$(0)=3$

y$(0)=2$