> restart
 

No initial conditions, so we get general solution (2 constants) 

> `:=`(ivp, {`+`(`*`(t, `*`(((`@@`(D, 2))(y))(t))), (D(y))(t), `-`(`*`(4, `*`(y(t)))))}); 1
 

{`+`(`*`(t, `*`(((`@@`(D, 2))(y))(t))), (D(y))(t), `-`(`*`(4, `*`(y(t)))))} (1)
 

> `:=`(Order, 5); -1; `:=`(ans1, dsolve(ivp, y(t), type = 'series'))
 

y(t) = `+`(`*`(_C1, `*`(series(`+`(1, `*`(4, `*`(t)), `*`(4, `*`(`^`(t, 2))), `*`(`/`(16, 9), `*`(`^`(t, 3))), `*`(`/`(4, 9), `*`(`^`(t, 4))))+O(`^`(t, 5)),t,5))), `*`(_C2, `*`(`+`(`*`(ln(t), `*`(seri...
y(t) = `+`(`*`(_C1, `*`(series(`+`(1, `*`(4, `*`(t)), `*`(4, `*`(`^`(t, 2))), `*`(`/`(16, 9), `*`(`^`(t, 3))), `*`(`/`(4, 9), `*`(`^`(t, 4))))+O(`^`(t, 5)),t,5))), `*`(_C2, `*`(`+`(`*`(ln(t), `*`(seri...
(2)
 

>
 

>
 

Now we add initial conditions to get a particular solution (no constants so we can graph) 

> `:=`(ivp, {`+`(`*`(t, `*`(((`@@`(D, 2))(y))(t))), (D(y))(t), `-`(`*`(4, `*`(y(t))))), y(1) = 1, (D(y))(1) = 0}); 1
 

{`+`(`*`(t, `*`(((`@@`(D, 2))(y))(t))), (D(y))(t), `-`(`*`(4, `*`(y(t))))), y(1) = 1, (D(y))(1) = 0} (3)
 

> `:=`(Order, 10); -1; `:=`(ans1, dsolve(ivp, y(t), type = 'series'))
 

y(t) = series(`+`(1, `*`(2, `*`(`^`(`+`(`-`(1), t), 2))), `-`(`*`(`/`(4, 3), `*`(`^`(`+`(`-`(1), t), 3)))), `*`(`/`(5, 3), `*`(`^`(`+`(`-`(1), t), 4))), `-`(`*`(`/`(8, 5), `*`(`^`(`+`(`-`(1), t), 5)))...
y(t) = series(`+`(1, `*`(2, `*`(`^`(`+`(`-`(1), t), 2))), `-`(`*`(`/`(4, 3), `*`(`^`(`+`(`-`(1), t), 3)))), `*`(`/`(5, 3), `*`(`^`(`+`(`-`(1), t), 4))), `-`(`*`(`/`(8, 5), `*`(`^`(`+`(`-`(1), t), 5)))...
(4)
 

Note: this series is based at t = 1 instead of t = 0 (the general solutions is at t = 0) 

> `:=`(ans2, convert(ans1, polynom))
 

y(t) = `+`(1, `*`(2, `*`(`^`(`+`(`-`(1), t), 2))), `-`(`*`(`/`(4, 3), `*`(`^`(`+`(`-`(1), t), 3)))), `*`(`/`(5, 3), `*`(`^`(`+`(`-`(1), t), 4))), `-`(`*`(`/`(8, 5), `*`(`^`(`+`(`-`(1), t), 5)))), `*`(...
y(t) = `+`(1, `*`(2, `*`(`^`(`+`(`-`(1), t), 2))), `-`(`*`(`/`(4, 3), `*`(`^`(`+`(`-`(1), t), 3)))), `*`(`/`(5, 3), `*`(`^`(`+`(`-`(1), t), 4))), `-`(`*`(`/`(8, 5), `*`(`^`(`+`(`-`(1), t), 5)))), `*`(...
(5)
 

> `:=`(ans3, op(2, ans2))
 

`+`(1, `*`(2, `*`(`^`(`+`(`-`(1), t), 2))), `-`(`*`(`/`(4, 3), `*`(`^`(`+`(`-`(1), t), 3)))), `*`(`/`(5, 3), `*`(`^`(`+`(`-`(1), t), 4))), `-`(`*`(`/`(8, 5), `*`(`^`(`+`(`-`(1), t), 5)))), `*`(`/`(14,...
`+`(1, `*`(2, `*`(`^`(`+`(`-`(1), t), 2))), `-`(`*`(`/`(4, 3), `*`(`^`(`+`(`-`(1), t), 3)))), `*`(`/`(5, 3), `*`(`^`(`+`(`-`(1), t), 4))), `-`(`*`(`/`(8, 5), `*`(`^`(`+`(`-`(1), t), 5)))), `*`(`/`(14,...
(6)
 

> `:=`(trange1, -4 .. 4); -1; `:=`(yrange1, -5 .. 5); -1; `:=`(p1, plot(ans3, t = trange1, color = blue)); -1
 

> `:=`(numer1, dsolve(ivp, type = numeric, range = trange1))
 

 

Warning, cannot evaluate the solution further left of .23262875e-152, probably a singularity
proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if `<`(1, nargs) then error (7)
 

> with(plots); -1; `:=`(p2, odeplot(numer1)); -1
 

> display(p2, p1, view = [trange1, yrange1])
 

Plot_2d
 

>