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Determine the exact solution to the initial value problem

 

y'(x)=   y(x)(20-y(x)) , y(0)=1

                 80

 

Compute a polynomial approximation to y(x). Plot this polynomial approximation together with y(x) on the same axes for x∈[0,20]. Choose different colours and linestyles for each curve.

 

Investigate whether or not it is possible to choose Order to be large enough to ensure that the plots of the polynomial approximation and y(x) are indistinguishable over the [0,20] interval? If this is possible, determine the minimum value of Order required. If you think that it is not possible, explain why not.

Hey, how is can i see all the steps in maple? I would specially like to know it for differential equations.

For example we could use this one:

dl := 3*(diff(y(t), t, t))+6*(diff(y(t), t))+4*y(t) = 0 

Hello,

 

could you help me solve this error ? I don't understand what it means.

 


> eq3:=diff(x(t),t,t)+Gamma*diff(x(t),t)+omega[0]^2*(x(t)-(diff(x(t),t,t)+Gamma*diff(x(t),t)+omega[0]^2*x(t)+omega[0]^2*X[0])/omega[0]^2) = -omega[0]^2*X[0]:
> dsolve(eq3);
Warning, it is required that the numerator of the given ODE depends on the highest derivative. Returning NULL.

 

Thanks.

Dear all,

I am trying to find the intial velocity of a ball that is shot under an angle while only the start and end coordinates are given. The air resistance should also be taken into account. 

In order to do that I have build the following Maple sheet:

Assignment_question_1.mw 

I have used two differential equations that both include the variables v0 and t, and then try to solve them. Only I receive an answer in the form of RootOf, which I cannot remove with for example allvalues. 

I have been working for quite a long time on this but I am not coming any further, so is there anyone who can find what I am doing wrong/what I should be doing else? Or maybe my whole approach is not right?

Even a small step in the right direction would be appreciated a lot!

Thanks in advance,

Elise

Please, how do i compute,solve and derive d' Alembert wave formula with maple since wave=How do i derive d'Alembert formula using maple and also how to solve any wave problems eg IC u(x,0)=1/2e^-x^2 and ut(x,0)-e^-x^2 when c=4

I want to get numerical solution of the Eqs.ode(see the folowlling ode and ibc)in Maple.However,when i run the following procedure,it prompts an error "Error, (in dsolve/numeric/bvp) cannot determine a suitable initial profile, please specify an approximate initial solution". How to solve the issue? Please help me.


restart:
n := 1.4; phi := 1; beta := .6931; psi := 1

> restart;
> n := 1.4; phi := 1; beta := .6931; psi := 1;

> s := proc (x) options operator, arrow; evalf(1+(phi*exp(beta*psi)*h(x))^n) end proc;

> Y := proc (x) options operator, arrow; evalf(f-(1/2-(1/2)/n)*ln(s(x))+2*ln(1-(1-s(x))^(-1+1/n))) end proc;


> ode := diff(h(x), `$`(x, 2))+(diff(Y(x), x))*(diff(h(x), x)+1) = 0;


> ibc := h(0) = 0, ((D(h))(10)+1)*s(10)^(-(1-1/n)*(1/2))*(1-(1-1/s(10))^(1-1/n))^2 = 0;

> p := dsolve({ibc, ode}, numeric);
Error, (in dsolve/numeric/bvp) cannot determine a suitable initial profile, please specify an approximate initial solution
>

solve Differential equation "a-y=y' bc" use maple 17

the result is y(x)=a+_C1e^-(1/bc)

but the correct result  isn't y(x)=a-_C1e^-(1/bc) ?

Thank you in advance for your help

 

please help me to find an analytical approach to the below equation:

> ode3 := diff(n(t), t)+(1/2)*(-(3.707186000*(0.815e-1*(diff(n(t), t, t))+diff(n(t), t)))/(0.815e-1*(diff(n(t), t))+n(t))^(3/2)-(.1428*(1+0.714e-1*n(t)))*(diff(n(t), t)))/sqrt(7.414372/sqrt(0.815e-1*(diff(n(t), t))+n(t))-(1+0.714e-1*n(t))^2)+n(t)+sqrt(7.414372/sqrt(0.815e-1*(diff(n(t), t))+n(t))-(1+0.714e-1*n(t))^2)-(2.518891688*(1+.3570*n(t)))*sqrt(0.815e-1*(diff(n(t), t))+n(t)) = 0;
                                                                           /
                                                                           |
/ d      \                                 1                               |
|--- n(t)| + ------------------------------------------------------------- |
\ dt     /                                                           (1/2) |
               /           7.414372                                2\      |
             2 |------------------------------- - (1 + 0.0714 n(t)) |      |
               |                          (1/2)                     |      \
               |/       / d      \       \                          |       
               ||0.0815 |--- n(t)| + n(t)|                          |       
               \\       \ dt     /       /                          /       
              /       / d  / d      \\   / d      \\
  3.707186000 |0.0815 |--- |--- n(t)|| + |--- n(t)||
              \       \ dt \ dt     //   \ dt     //
- --------------------------------------------------
                                     (3/2)          
           /       / d      \       \               
           |0.0815 |--- n(t)| + n(t)|               
           \       \ dt     /       /               

                                        \       
                                        |       
                              / d      \|       
   - 0.1428 (1 + 0.0714 n(t)) |--- n(t)|| + n(t)
                              \ dt     /|       
                                        |       
                                        |       
                                        /       

                                                           (1/2)
     /           7.414372                                2\     
   + |------------------------------- - (1 + 0.0714 n(t)) |     
     |                          (1/2)                     |     
     |/       / d      \       \                          |     
     ||0.0815 |--- n(t)| + n(t)|                          |     
     \\       \ dt     /       /                          /     

                                                             (1/2)    
                                   /       / d      \       \         
   - 2.518891688 (1 + 0.3570 n(t)) |0.0815 |--- n(t)| + n(t)|      = 0
                                   \       \ dt     /       /         
> ics := n(0) = 0, (D(n))(0) = 674.5142595;


thanks and regards

louiza

 

AOA... Dears! When i solve the following differential equations

-(diff(lambda(s), s))-2*(diff(lambda(s), s, s))-(diff(lambda(s), s, s, s)) = 0

i got

lambda(s) = _C1+_C2*exp(-s)+_C3*exp(-s)*s

 here _C1,_C2 and _C3 are constant of intergration but i want the constant of integration of the following type

C[1],C[2] and C[3]

due to some reson pl help

Hi, I have a homework to do that I am strugling with:

write a procedure which uses euler's method to solve a given initial value problem.
the imput should be the differential equation and the initial value.
using this programme find y(1) if dy/dx= x^2*y^3 and y(0)=1, and use maple dsolve command to check the solution.

That is what I have managed to do, but somehow it is not working correctelly, can somebody help please?

eul:=proc(f,h,x0,y0,xn)
  local no_points,x_old,x_new,y_old,y_new,i:
  no_points:=round(evalf((xn-x0)/h)):
  x_old:=x0:
  y_old:=y0:
 
  for i from 1 to no_points do
      x_new:=x_old+h:
      y_new:=y_old+evalf(h*f(x_old,y_old)):
      x_old:=x_new:
      y_old:=y_new:
  od:
  y_new:
end:


Thanks

I'm trying to plot the direction field of the second order differential equation x''=x'-cos(x) using dfieldplot: 

> with(DEtools); with(plots);
> f1 := (x, y) options operator, arrow; diff(x(t), t)-cos(x(t)) end proc;
/ d \
(x, y) -> |--- x(t)| - cos(x(t))
\ dt /
> dfieldplot([diff(x(t), t) = y(t), diff(y(t), t) = f1(x(t), y(t))], [x(t), y(t)], t = -2 .. 2, x = -2 .. 2, y = -2 .. 2);
Error, (in DEtools/dfieldplot) cannot produce plot, non-autonomous DE(s) require initial conditions.
>

The error I'm getting says I need initial conditions, but I wasn't provided with any. Is there another way to plot this? Sorry if this is dumb question, but I've only ever plotted first order equations.

I have to solve a system composed of a mass, a spring and a damper, represented by this equation :

m (d2x/dt2) + c (dx/dt) + k x(t) = F(t)

with m the mass, t the time, c the constant of the damper, k the constant of the spring, F an external force applied to the mass and x(t) the movement of the mass m at time t.

Please help me to solve this equation on Maple.

What is wrong???

restart;
with(DEtools); with(plots);
epsilon := 'epsilon';

epsilon := 0.3e-1;
h := .75;
p := 2;
q := 0.6e-2;

sol := dsolve([[epsilon*(diff(x(t), t)) = x(t)+y(t)-q*x(t)^2-x(t)*y(t), diff(y(t), t) = h*z(t)-y(t)-x(t)*y(t), p*(diff(z(t), t)) = x(t)-z(t)], [x(0) = 100, y(0) = 1, z(0) = 10]], type = numeric);
%;
Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

I write this system but I have 2 error

 

restart; params := [z = 0,

Omega = 2.2758,

tau = 13.8, T2 = 200,

omega0 = 1,

r = .7071,

s = 2.2758,

omega = .5]

 

sys1 := {diff(q(t), t) = -2*Omega*v(t)-s*exp(-r^2/omega0^2-t^2*1.177^2/tau^2)*cos(k*z-omega*t)*(y(t)-x(t))-q(t)/T2,

diff(v(t), t) = Omega*q(t)-v(t)/T2,

diff(x(t), t) = 2*s*exp(-r^2/omega0^2-t^2*1.177^2/tau^2)*cos(k*z-omega*t)*q(t)+y(t)/T1,

diff(y(t), t) = -2*s*exp(-r^2/omega0^2-t^2*1.177^2/tau^2)*cos(k*z-omega*t)*q(t)-y(t)/T1};

ICs1 := {q(-20) = 0, v(-20) = 0, x(-20) = 1, y(-20) = 0}

 

 

ans1 := dsolve(`union`(eval(sys1, params), ICs1), numeric, output = listprocedure); plots:-odeplot(ans1, [[t, x(t)], [t, y(t)], [t, q(t)], [t, v(t)]], t = -20 .. 20, legend = [x, y, q, v])

 

Error, invalid input: eval received params, which is not valid for its 2nd argument, eqns
Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

http://homepages.lboro.ac.uk/~makk/MathRev_Lie.pdf

ode1 := Diff(f(x),x$2)+2*Diff(f(x),x)+f(x);
with(DEtools):
with(PDETools):
gen1 := symgen(ode1);
with(PDEtools):
DepVars := ([f])(t);
NewVars := ([g])(r);
SymmetryTransformation(gen1, DepVars, NewVars);

Error, invalid input: too many and/or wrong type of arguments passed to PDEtools:-SymmetryTransformation; first unused argument is [_xi = -x, _eta = f*x]


generator1 := rhs(sym1[3][1])*Diff(g, x)+ rhs(sym1[3][2])*Diff(g, b)

what is X1 and X2 so that [X1, X2] = X1*X2 - X2*X1 = (X1(e2)-X2(e1))*Diff(g, z) ?

is it possible to use lie group to represent a differential equation, and convert this group back to differential equation ? how do it do?

 

how to find symmetry z + 2*t*a, when you do not know before taylor calcaulation?

fza := z + 2*t*a;
fza := x;
fza := z + subs(a=0, diff(fza,a))*a;

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