## 1401 Reputation

7 years, 165 days

## Gamma......

This integral, in most cases, cannot be expressed in terms of elementary functions,but we can expressed in terms of GAMMA function.

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 Int(exp(beta2*_z1)*(1+N-_z1)^(-theta),_z1 = T .. t) = -1/beta2*exp(N*beta2+ beta2)*((1+N-t)^(-theta)*(beta2*(1+N-t))^theta*(theta*GAMMA(-theta)+GAMMA(1- theta,beta2*(1+N-t)))+(1+N-T)^(-theta)*(beta2*(1+N-T))^theta*(GAMMA(1-theta)- GAMMA(1-theta,beta2*(1+N-T))))

```
```

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## Try:...

Do you have any reason to think there is a closed form?

Most integrals don't have one.

Maybe the best you can do is numerical methods.

See atthached file.

ode.mw

See teory of First Order Differential Equations. Only one initial value problem can be not two.

## Try:...

See attached file:

integral.mw

See attached file:

PDE_by_Elziki_Transform.mw

## Homework.....

For first question:

```f := x -> 36*x^6 + 2665*x^4 + 240*x - 675 + 4534*x^2 - 5836*x^3 - 516*x^5;

minimize(f(x), x = 0 .. 4, location);
#-675, {[{x = 0}, -675]}

evalf(maximize(f(x), x = 0 .. 4, location));
#703.9550742, {[{x = 3.800387934}, 703.9550742]}```

## Try...

Try:

```ode := diff(U(z), z \$ 4) + c^2*diff(U(z), z \$ 2) + k*c*diff(U(z), z \$ 2) - (3*U(z)^2 + a)*diff(U(z), z \$ 2) = 0;
Order := 5;dsolve(ode, U(z), type = 'series');

#U(z) = U(0) + D(U)(0)*z + 1/2*(D@@2)(U)(0)*z^2 + 1/6*(D@@3)(U)(0)*z^3 + (U(0)^2*(D@@2)(U)(0)/8 - c^2*(D@@2)(U)(0)/24 - k*c*(D@@2)(U)(0)/24 + (D@@2)(U)(0)*a/24)*z^4 + O(z^5)```

With initial conditions

```Order := 5;dsolve([ode, U(A) = A1, D(U)(A) = B1, (D@@2)(U)(A) = C1], U(z), type = 'series');

#U(z) = A1 + B1*(z - A) + 1/2*C1*(z - A)^2 + 1/6*(D@@3)(U)(A)*(z - A)^3 + (1/8*A1^2*C1 - 1/24*c^2*C1 - 1/24*k*c*C1 + 1/24*C1*a)*(z - A)^4 + O((z - A)^5)```

## Workaround....

As a workround using fourier transform:

(inttrans:-invfourier(int((inttrans:-fourier(sin(p*r), p, s) assuming (0 <= r))*sin(q*r)/(p*q), r = 0 .. infinity), s, p) assuming (q < p));

#-Pi*Dirac(p + q)/(2*p*q)

## Try: simplify(pdetest(sol, sys)); giv...

Try:

`simplify(pdetest(sol, sys));`

gives:

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

## General formula for n...

Maybe this helps, see attached file:

General_formula_.mw

## Or try:...

```simplify(diff(int(JacobiSN(x, k)^2, x), x));
```

#JacobiSN(x, k)^2

See attached file:

EQ_v3.mw

## Only......

Only solution,not  phase portrait.See atached file.

Solution.mw

## A way:...

One way is:

`[seq(rhs(op(1, rootsq0[[n]])), n = 1 .. numelems([rootsq0]))];`

## Maybe...

Maybe like this:

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/integrals.mw .