My explanation: By default, plot uses Hardwarefloats and cannot evaluate such large numbers:

E[1]:=-3.27*10^(-10)*(-6.02*10^78-6.18*10^78*v)^2*(-9.23*10^232*v-1.15*10^232-9.03*10^232*v^3-1.74*10^233*v^2)*(-7.91*10^154-1.94*10^155*v-1.11*10^155*v^2)/(-4.31*10^534*v-3.11*10^535*v^4-3.13*10^535*v^3-3.90*10^533-3.28*10^534*v^6-1.59*10^535*v^5-1.66*10^535*v^2);

p1:=plot(E[1],v=0..0.5):plottools[getdata](p1)[3][1..10]

If you raise Digits, it switches to soft data types and it works without simplification: 
Digits:=16;p1:=plot(E[1],v=0..0.5);

You may try to reduce screen resolution

Instead of debugging or tracing commands, you may use the teaching options of Maple.

with(Student[Calculus1]):
ShowSolution(Limit(x^2 *log(x),x=0));
 

Although in this particular case, the step of showing l'Hospital's rule is not well illustrated due to automatic simplification. See ?student,calculus1 for more single step tasks relating calculus. 

Here you go:

f:=(r,n)->r*(sin(n*2*Pi/12),cos(n*2*Pi/12)),n;
s:=seq([f(1,n)],n=1..12);
plots[textplot]([s],axes=none);

Here's one approach. In Maple, It is hard to do term manipulations exactly as you want. Often it is trial and error. 

restart;
ode:=diff(y(x),x) = sec(x)^2*sec(y(x))^3;
sol:=dsolve(ode, y(x),implicit);
expand(sol):collect(%,sin);
 

Not sure where this numeric exception comes from, but we can find a partial solution with

PDEtools:-casesplit(x*(a^2*x+(x^2-y(x)^2)*y(x))*diff(y(x),x)^2-(2*a^2*x*y(x)-(x^2-y(x)^2)^2)*diff(y(x),x)+a^2*y(x)^2-x*y(x)*(x^2-y(x)^2) = 0);
ode1:=diff(y(x), x) = y(x)/x; sol1:=dsolve(ode1,y(x));
odetest(sol1,ode1);
 

One way would be to do a laplace transform, but I think in this engineering context, the DynamicSystems package is better here, it lets you interactively setting parameters and doing conversions. By the way, your second equation is not correct, must be Vi=R1.I1+R2.I2+V2

Once you have entered your system and the package loaded, the context menu (at the bottom of list) gives you various options

Transfer_Function.mw

Edit: I see you are using Maple 7. Seriously, you are missing alot with this 16 year old version. You probably have to try laplace transform then.

try something like

simplify(%,size) assuming n::even

It would really help if you would post executable code instead of a picture

you need to place assumptions on y, like assume(y>0);

you still get an integral that probably cannot be solved explicitly

I guess you just have to write the integral so that you can use it like myJ(1.2) and get a numeric answer.

So, write myJ:=x->1/Pi*int...   and so on

u:=(3*x-1)*(1-x):
v:=2*x*(1-2*x):

plot([u, v, x = 0 .. 1], scaling = unconstrained, gridlines = true)

Have a look at ?StringTools,DifferencePositions

using first integrals:

restart; with(DEtools):
dgl := diff(u(x), x, x) = a__0+a__1*u(x)+a__2*u(x)^2;
mu := intfactor(dgl);
firint(mu*dgl);

What do you mean by fractional differential equation? It is a common ODE you can solve with  dsolve

R*(diff(q(t), t))+q(t)/C = v(t);
dsolve(%);

 use intfactor:

ode := omega*x+sigma*y(x)+(omega*y(x)-sigma*x)*(diff(y(x), x)) = 0;
mu := DEtools[intfactor](ode);

#test:
DEtools[odeadvisor](ode);
DEtools[odeadvisor](mu*ode);