MATH528 Lesson05
Homogeneous, constant coefficient, second-order ODEs

General solution

ODE=y''[x]+a y'[x]+b y[x]==0;iCond1=y[0]==y0;iCond2=y'[0]==y0p;
sol[x_] = y[x] /. 
DSolve[{ODE,iCond1,iCond2} ,y[x],x][[1,1]]

ToString[1.2]

Manipulate[Plot[sol[x] /. {a->av, b->bv, y0->y0v,y0p->y0pv},{x,0,100},PlotRange->{{0,100},{-5,5}},Axes->False,Frame->True,GridLines->Automatic, FrameLabel->{"x","y(x)"},PlotLabel->"(Subscript[λ, 1])=" <>ToString[(1/2) (-av-(Sqrt[(av^2)-4 bv]))]<> " (Subscript[λ, 2])=" <> ToString[(1/2) (-av+(Sqrt[(av^2)-4 bv]))]],{{av,1},-1,1},{{bv,0},-1,1},{{y0v,0},-1,1},{{y0pv,1},-1,1}]