For the Case III use: 

dsolve(eqs, [p(t), y(t)])

a := solve(eq, [x]);allvalues(a);

@adam25185 

For me works fine.

Download integral.mw

Which integral is wrong ?

@Alfred_F 

See:

Download eq.mw

@Alfred_F 

If "text is difficult to read" clik right mouse button on picture and open image in new tab, then should work.

infolevel[isolve] := 5;

infolevel[solve] := 5;

isolve(x^2 - 12*x*y + 6*y^2 + 4*x + 12*y - 3)

#isolve: Warning: unable to find solution over the integers.

(solve(x^2 - 12*x*y + 6*y^2 + 4*x + 12*y - 3, {x, y}, useassumptions = true) assuming (x in integer, y in integer))

#Warning, solve may not respect assumed property 'integer' on 'x'.
#Warning, solve may not respect assumed property 'integer' on 'y'.

#{x = 6*y - 2 + sqrt(30*y^2 - 36*y + 7), y = y}, {x = 6*y - 2 - sqrt(30*y^2 - 36*y + 7), y = y}

Mathematica gives:

The integral diverges !!!

help("piecewise");

and "Enter" ?

@Dkunb 

I doubt  be  can solvelable in Maple even near future,unless the people from Maple they will get to hard work.

Even Mathematica can't solve.

@Joseph Poveromo 

In my opinion it should be: int(y'(x)*x^2/(x^2-1),x) = int(sqrt(y(x)),x)^(-2/3)

 int[y'(x)* (x^2)/[(x^2)-1],x)  =  (int[sqrt(y(x)])^(-2/3)

Maple give me errors ?

@Prakash J 

Use:

dsn := dsolve([eq1, eq2, eq3, eq4, ics], numeric, delaymax = tau, method = ck45);
dsn := dsolve([eq1, eq2, eq3, eq4, ics], numeric, delaymax = tau, method = rkf45);

From Help:

"rkf45 finds a numerical solution using a Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant. This is the default method of the type=numeric solution for initial value problems when the stiff argument is not used.  The other non-stiff method is a Runge-Kutta method with the Cash-Karp coefficients, ck45.
"

@yaseentareq 

restart;
InverseElzakiTransform := proc(F, v) inttrans:-invlaplace(eval(F/v, v = 1/v), v, t); end proc;
F := v^(n + 2);
(InverseElzakiTransform(F, v) assuming (0 < n));

#t^n/GAMMA(1 + n)

If inverse laplace transform can't solve we have to use another way or another mathematical mainpulation.

InverseElzakiTransformBYInverseFourier := proc(F, v) (limit(exp(-Z*t)*inttrans:-invfourier(eval(eval(F/v, v = 1/v), v = Z + w*I), w, t), Z = 0, right) assuming (0 < t)); end proc;
F := v^3/(v^2 + 1);
InverseElzakiTransformBYInverseFourier(F, v);

#sin(t)

InverseElzakiTransformBYInverseMellin := proc(F, v) (simplify(eval(inttrans:-invmellin(eval(F/v, v = 1/v), v, t, 1 .. infinity), t = exp(-t))) assuming (0 < t)); end proc;
F := v^3/(v^2 + 1);
(InverseElzakiTransformBYInverseMellin(F, v) assuming (0 < n));

#sin(t)

I can only write in Maple Integral Transform are very Very weak.

Please share the code in copyable form, so that other users can play with it. No one wants to retype all this code (and double check for correct transition). This will raise your chances for getting quick and competent help.