1) [8] (2) Use Euler's Method to approximate the solution to I'(!) = 1+ isin(ux), x(0) =0 at +=1 using ten steps. (b) Do the same with the improved Euler method and compare results. 1) [7] Solve the initial value problem y'+ 2y+ y=0. )(0)=1.y(0)=-2. 3) Solve the initial value problem tan(y) -2+(xsec (pit - '(x) =0, y(0) =1. 4) [8] Find the general solution for the differential equation: y"(x) - 5y'(x) + 4y(x) = Ber 5) [8] Solve the initial value problem y"(x) + 4y'(x) + 3y(x) =0;y(0) = 1:y'(0) = 3. 6) [8] Consider the initial value problem: 2y" +xy'ty=0, y(0) = 0, y'(0) =1 Use a power series expansion about x=0 to find a general solution for the problem by determining at least the first five nonzero terms. Plot the resulting polynomial on (-10.10).7) [8] A damped vibrating spring under an external driving force can be modeled by the equation my"thy'thy =p(0 where m:30 is the mass of the spring. & is the damping constant, *30 the spring constant and (t) the driving force. If)() is the displacement from equilibrium at time r, determine the form (general solution) of the equation of motion if gor) = sin(lr) (assume b)