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7 years, 165 days

Symbolic solution....

I don't have Maple 18.I don't understand what you mean by: h^k (0)=0 for k=0..n  ?

Solution by LaplaceTransform:

See attached file:

Download frac_eq.mw

frac_eq_2.mw

odetest...

```sol := dsolve((D@@2)(z)(t) + 2*D(z)(t) + z(t) = 2*exp(-t));
odetest(sol, (D@@2)(z)(t) + 2*D(z)(t) + z(t) = 2*exp(-t));
#0 ok.
```

In your case:

```odetest(z(t) = t^2 + exp(-t), (D@@2)(z)(t) + 2*D(z)(t) + z(t) = 2*exp(-t));
#2 - 2*exp(-t) + 4*t + t^2```

is not true.

For more info execute ?odetest.

Another way:...

```genfunc:-rgf_pfrac(1/(z^2 + I)^2, z);

#(-1/4 + I/4)*sqrt(2)/(-sqrt(2)*I + sqrt(2) + 2*z) + I/(-sqrt(2)*I + sqrt(2) + 2*z)^2 + I/(sqrt(2)*I - sqrt(2) + 2*z)^2 + (1/4 - I/4)*sqrt(2)/(sqrt(2)*I - sqrt(2) + 2*z)```

plot...

Using LaplaceTransform and Inverse LaplaceTransform. Fractional derivative is Riemann–Liouville sense.

```restart;
v := t -> t;
plot([seq((inttrans:-invlaplace(s^alpha*inttrans:-laplace(v(t), t, s), s, t) assuming (0 < t)), alpha = -1 .. 1.5, 0.5)], t = 0 .. 10, view = [0 .. 9.5, 0 .. 5], legend = [seq('alpha' = alpha, alpha = -1 .. 1.5, 0.5)], color = [red, blue, green, yellow, cyan, magenta], axis[2] = [gridlines = [linestyle = dot]]);```

Maybe...My answers helps...

Elzaki transform by Laplace Transform, see attached file.

Elziki_transforms_vs_2.mw

Maybe......

Maybe:

```func1 := h(x) = 2/f(x);
func2 := diff(func1, x);
subs([diff(h(x), x) = D(h)(-1), diff(f(x), x) = 2, f(x) = 4], func2);```

#h'(-1) = -1/4

A little correction....

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Download odeplot_test_ver2.mw

Only one solution....

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Download pde_ver2.mw

Workaround:...

I don't know why Maple 2020.1 can't give a solution,probably a weakness in here.

All computer algebra systems, including Maple, are limited in their capabilities.

The delta function can be defined as the following :

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I tried with Mathematica 12.1.1 and works fine,give me 3.

As you can see here Mathematica(Rubi) is a leader for solving a indefine integrals probably also for define integrals.

Download dirac_integral.mwDownload dirac_integral.mw

Try...

Try:

```restart;

f:=x->x^5+2*x^3+3;

[f(x), f(1), f(-1)];```

Output: [x^5 + 2*x^3 + 3, 6, 0]

For me is ok....

pdsolve give you a correct solution see here.

pdetest for some reason(is not smart  enough) to check if is true,this does't mean that pdetest gave you the wrong solution.

IBVP_ver2.mw

Maybe:...

Maybe so:

plot(-0.25*t^2 + 16.9*t - 50, t = 5 .. 29, tickmarks = [[5 = "1995", 10 = "2001", 15 = "2007", 20 = "2013", 25 = "2019", 29 = "2024"], default]);

or:

plot(-0.25*t^2 + 16.9*t - 50, t = 5 .. 29, tickmarks = [[seq(n = convert(1988.9583333333333333 + 1.2083333333333333333*n, rational, 4), n = 5 .. 29, 4)], default]);

Analytic solution...

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Download solution.mw

Select for Palletes -> Comon Sy...

Select for Palletes -> Common Symbols -> imaginaryunit:  I (uppercase), or (lowercase).

Example 2: Multiplying Complex Numbers

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Download Multiplying_Complex_num_ver2.mw

No closed-form probably exists....

I think that your integral  does not have finite closed-form expression in terms of very large class of special functions.

Most integrals don't have one. I have only approximation by infinite Sum.

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Download integral.mw

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