Can someone help me with the following question?

I would like to plot the function f(x)=a*x*(1-x)+x*ln(x)+(1-x)ln(1-x) in the interval [x1,x2], where x1 and x2 are points that satisfy the following: df/dx(x=x1)=df/dx(x=x2)=0 and f(x1)=f(x2), where a is some parameter that is greater than 0.

I guess I need to use here fsolve, but I am not sure how.

I have the following simple dsolve code:

ode := diff(y(x), x, x) + (n*pi)^2*y(x) = A^3*sin(n*pi*x)^3;
dsol1 := dsolve({ode, y(0) = 0, y(1) = 0}, y(x));

How to declare that n is an integer here?

Thanks!

I have this polynomial equation: (x-2)^2*(x-3)+epsilon =0, I want to draw a bifurcation diagram in the (epsilon , x) plane.

How to implement this in maple 2018?

Thanks!

I have the following ODE perturbation problem which I want maple to solve for me:

q'(\tau)=f(p(eps*\tau)+eps*q(\tau),r(eps*\tau)+s(\tau))-f(p(eps*\tau,r(eps*\tau)+s(\tau))-f(p(eps*\tau),r(eps*\tau))

where q(\tau)=q_0(\tau)+eps*q_1(\tau)

p(eps*\tau)=p_0(eps*\tau)+eps*p_1(eps*\tau)

s(\tau)=s_0(\tau)+eps*s_1(\tau)

r(eps*\tau)=r_0(eps*\tau)+eps*r_1(eps*\tau)

I want maple to expand every function that depends on eps in its arguments by a Taylor series around eps=0, i.e h(eps)=h(0)+eps*h'(0)

and also expand the difference above the fs with an eps-expansion around eps=0.

I did all this manually now I want to check if my calculations are correct, eventaully I want to equate same powers of eps of the RHS and LHS of the first ODE I wrote above.

Then how to use maple for this?

Thnaks.

I want to compute a limit via maple and that it will show me the way how to compute the limit.

The limit is:

\lim_{epsilon ->0, t\in [0,1]} 1/(exp((-1+(1-4*epsilon)^(0.5))/(2*epsilon))-exp((-1-(1-4*epsilon)^(0.5))/(2*epsilon)))*[exp((-1+(1-4*epsilon)^(0.5))/(2*epsilon)*t)-exp((-1-(1-4*epsilon)^(0.5))/(2*epsilon)*t)]/(exp(1-t)-exp(1-t/(epsilon)))

According to my book it should converge to 1.

I tried manually but got stuck.