Use the vectorized Euler Method with h=0.5 to find an approximation for the solution to the given initial value problem on the specialized interval. y''+ty'+4y=0; y(0)=1, y'(0)=0 on [0,2]

4. Consider the motion of a particle as described by 1 1 r(t) = = t, Write the acceleration vector of this particle at time t = 1 as a = a+T+ anÑ. You do not need to find T and Ñ, but you can if you find it helpful.

a.) Form the complimentary solution to the homogeneous equation. y_c(t) = c_1 [ _ _] + c_2 [ _ _] b.) Construct a particular solution by assuming the form y_P(t) = e^(-3t)a and solving for the undetermined constant vector a. y_P(t) = [ _ _] c.) Form the general solution y(t) = y_c(t) + y_P(t) and impose the initial condition to obtain the solution of the initial value problem. [y_1(t) y_2(t)] = [ _ _]

Use the method of variation of parameters to produce a general solution of the inhomogenous equation y" + 2y' + y = x² +1