Using Differential Equations Made for the TI89 we enter the 3 problems as shown below. Screenshots show how the problems were entered and the solutions.

Question 2) y”-2y+5y=0 y(0)=2 y'(0)=4

go to F2 2 and enter b=-2 and c=5 , x0=0, y(x0)=2 y,(x0)=4

Question 3) y”-3y+2y=6e^(-x) y(0)=1 y'(0)=2 .

go to F2 3 and enter b=-3 c=2 , g=6e^(-x) x0=0 y(x0)=1 y'(x0)=2 . The corresponding homogeneous solution can be found with steps under F2 2

Question 4) y”’-4y”+13y’=e^(-x)

the solution is the sum of the homogeneous and particular solutions. Thus, use 2 steps.

first find the homogeneous solution under F3 1 and enter t^3-4t^2+13t=0

next F3 2 . a=-4 b=13 c=0 g=e^(-x) to find part. solution -1/18 e^(-x)

Just add the two solutions.