Solving ODE with Laplace
>ode &= 'diff(y(t),t,2)+5*'diff(y(t),t)+4*y(t)=t
2
d d
--- (y(t)) + 5 (-- (y(t))) + 4 y(t) = t
2 dt
dt
>&atvalue(y(t),t=0,0);
>&atvalue('diff(y(t),t),t=0,0);
>&laplace(ode,t,s), sol &= solve(%,'laplace(y(t),t,s))
2
s laplace(y(t), t, s) + 5 s laplace(y(t), t, s)
1
+ 4 laplace(y(t), t, s) = --
2
s
1
[laplace(y(t), t, s) = ----------------]
4 3 2
s + 5 s + 4 s
>function ysol(t) &= ilt(rhs(sol[1]),s,t)
- t - 4 t
E E t 5
---- - ------ + - - --
3 48 4 16
>plot2d("ysol",0,1); insimg;

>ode &= 'diff(y,t,2)+5*'diff(y,t)+4*y=t
2
d y dy
--- + 5 -- + 4 y = t
2 dt
dt
>sol &= ode2(ode,y,t)
- t - 4 t 4 t - 5
y = %k1 E + %k2 E + -------
16
>&ic2(sol,t=0,y=0,'diff(y,t)=0)
- t - 4 t
E E 4 t - 5
y = ---- - ------ + -------
3 48 16
>&solve([at(rhs(sol),t=0)=0,diffat(rhs(sol),t=0)=0],[%k1,%k2]), &sol with %[1]
1 1
[[%k1 = -, %k2 = - --]]
3 48
- t - 4 t
E E 4 t - 5
y = ---- - ------ + -------
3 48 16

>&solve(h^2+5*h+4=0)
[h = - 4, h = - 1]


>yab &= a*t+b
a t + b
>&diff(yab,t,2)+5*diff(yab,t)+4*yab|ratsimp, ...
4 a t + 4 b + 5 a
> &solve([coeff(%,t,0)=0,coeff(%,t,1)=1],[a,b])
1 5
[[a = -, b = - --]]
4 16
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