function y=lapode22(a,b,c,q1,y0,yprime0)
% - Solving the second ode a y''+by'+cy=q1(x) with y(0)=y0 y'(0)=yprime0
% - by laplace transform.
% - Let       F= laplace(sym('y'))
% -      then    laplace(diff(sym('y'))= sF-y0
% -              laplace(diff(sym('y'),x,2)=s^2*F-s*y0-yprime0.
% -
% -  furthermore let phi=laplace(q1)
% - so  laplace of {  ay''+by'+cy=q1  } translates into an algebraic equation:
% -      F=   ( phi +(a*s+b)*y0+yprime0)/(a*s^2+b*s+c)
% - so   y=inverse laplace transform of F
% -  return  y=ilaplace(F)
% -  usage syms x define  a,b,b,q1(x),y0,yprime0 and y=lapode22(a,b,c,q1,y0,yprime0)
syms x s phi 
phi=laplace(q1)
F=simplify((phi+(a*s+b)*y0+yprime0)/(a*s^2+b*s+c))
F=factor(F)
y=ilaplace(F);
t=x;
y=subs(y);
return