Mariusz Iwaniuk

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These are answers submitted by Mariusz Iwaniuk

solve(15-(1/100)*m = 5+(1/600)*m, {m})
#{m = 6000/7}

solve(15-1/(100*m) = 5+1/(600*m), {m})
#{m = 7/6000}

Regards,Mariusz

Please read in Maple Help (Ctrl+F1 and put "How Do I,Solve an Ordinary Differential Equation?" in Search->Enter):

Scroll down to:

      -Solving an ODE Numerically

      - Taking Derivatives and Integrals of Numeric Solutions

      -  Why You Should Not Use Int or Diff on a Numeric Solution 

 

Corrected file:question-vers_2.mw

(Edited: 2 times) :P

    
 


 

restart

with(Student[Calculus1])

solve([-(1.25*y-sqrt(abs(x)))*abs(1, x)/sqrt(abs(x))+2*x, 3.1250*y-2.50*sqrt(abs(x))], [x, y])

[]

(1)

solX := solve(((125*(1/100))*y-sqrt(abs(x)))^2+x^2-1 = 0, {y})

{y = (4/5)*abs(x)^(1/2)+(4/5)*(-x^2+1)^(1/2)}, {y = (4/5)*abs(x)^(1/2)-(4/5)*(-x^2+1)^(1/2)}

(2)

c1 := CriticalPoints(rhs(solX[1, 1]), x)

[-1, -(1/12)*((215+12*321^(1/2))^(2/3)-(215+12*321^(1/2))^(1/3)+1)/(215+12*321^(1/2))^(1/3), 0, (1/12)*((215+12*321^(1/2))^(2/3)-(215+12*321^(1/2))^(1/3)+1)/(215+12*321^(1/2))^(1/3), 1]

(3)

evalf(c1)

[-1., -.5566930951, 0., .5566930951, 1.]

(4)

point1 := evalf(subs(x = 1, rhs(solX[1, 1])))

.8000000000

(5)

point2 := evalf(subs(x = 0, rhs(solX[2, 1])))

-.8000000000

(6)

point3 := evalf(subs(x = -.5566930951, rhs(solX[1, 1])))

1.261469543

(7)

c2 := CriticalPoints(rhs(solX[2, 1]), x)

[-1, 0, 1]

(8)

with(plots); p1 := pointplot([[evalf(c1)[2], point3], [evalf(c1)[4], point3], [0, point2], [0, point1]], color = [green], symbol = circle, axes = none, symbolsize = 15); p2 := implicitplot(((125*(1/100))*y-sqrt(abs(x)))^2+x^2-1, x = -1 .. 1, y = -1 .. 1.3, axes = normal, gridrefine = 3); display({p1, p2})

 

``


 

Download Critical_Points.mw

sol := n > ceil(evalf(solve(product(exp(1/i), i = 1 .. n) = 100, n))); solve(sol, n);

# sol := 56 < n

# RealRange(Open(56), infinity)

Calculating point with newton method:

restart;
with(Student[NumericalAnalysis]):
with(plots):
f := x^3-x^2-x-1;
P := Newton(f, x = 2.0, tolerance = 10^(-2));
p1 := pointplot([[P, 0]], color = [blue], symbolsize = 20, symbol = circle, axes = normal):
p2 := plot(f, x = 0 .. 2.2):
display({p1, p2});

 

First iteration:   

evalf(eval(simplify(x-f/(diff(f, x))), x = 2));
# 1.857142857

 

Hi

I'm only changed "Equation" to "2*M" , "Solu" to "Solu[1]" and other things.

Regards Mariusz.

Help_v2.mw

 

 I'm not an expert on that topic.

With help from: https://www.maplesoft.com/applications/view.aspx?sid=4971  (See: A Numeric Approach).

 

restart;
f := proc (u) options operator, arrow; piecewise(0 <= u and u <= 1, 0, 1 < u and u <= 2, 1) end proc;
ode := diff(y(u), u$2) = 4*Pi^2*(f(u)-e)*y(u); bc := y(0) = 0, y(2) = 0, (D(y))(0) = 1;
Eigen1 := (dsolve({bc, ode}, numeric, range = 0 .. 2, maxmesh = 8192, abserr = 1.*10^(-3), approxsoln = [y(u) = exp(-u), e = 1]))(0)[4];
Eigen2 := (dsolve({bc, ode}, numeric, range = 0 .. 2, maxmesh = 8192, abserr = 1.*10^(-3), approxsoln = [y(u) = u, e = 3]))(0)[4];
Eigen3 := (dsolve({bc, ode}, numeric, range = 0 .. 2, maxmesh = 8192, abserr = 1.*10^(-3), approxsoln = [y(u) = u, e = 6]))(0)[4];

In Maple there is not much choice how to do it.

One of the possibilities is textplot.

with(plots):
p1 := pointplot([[3, 8], [-5, 16], [11, 32], [3, -8]], color = [black], symbol = solidbox, axes = none, symbolsize = 12): 
p2 := textplot({[-5, 16+2, "lampart"], [3, (-8)-2, "dog"], [3, 8+3, "cat"], [11, 32+4, "panthera"]}, axes = none):
p3 := implicitplot(y^2 = x^3-43*x+166, x = -40 .. 40, y = -40 .. 40, axes = normal, gridrefine = 2): 
display({p1, p2, p3})

Your formula is very complicated,closed form solution may be not exist(Mathematica also can't solve)

Maybe you can try a Numeric Inverse Laplace to solve your problem.

Help.mw

 


 

X := [seq(i, i = 0 .. 24)]; Y := [1154, 1156, 1156, 1155, 1152, 1143, 1105, 1069, 1051, 1077, 1117, 1154, 1154, 1156, 1158, 1157, 1155, 1152, 1128, 1089, 1058, 1059, 1092, 1130, 1163]

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]

 

[1154, 1156, 1156, 1155, 1152, 1143, 1105, 1069, 1051, 1077, 1117, 1154, 1154, 1156, 1158, 1157, 1155, 1152, 1128, 1089, 1058, 1059, 1092, 1130, 1163]

(1)

with(Statistics)

f := Fit(a*sin(b*x+c)+d, X, Y, x, initialvalues = [a = 50, b = -1/2, c = 2, d = 1100])

-HFloat(48.70292499204853)*sin(HFloat(0.5173979580818461)*x-HFloat(2.5826651644523233))+HFloat(1124.950148688867)

(2)

plot({f(x), [seq([X[i], Y[i]], i = 1 .. 25)]}, x = 0 .. 24, style = [line, point])

 

``


 

Download fit.mw

Using another initalvalues:

f := Fit(a*sin(b*x+c)+d, X, Y, x, initialvalues = [a = 50, b = 1/2, c = 1/2, d = 1120]);

#f := 48.7029442510649*sin(0.517394221899901*x+0.558920097492524)+1124.95011131587

Maple gives almost the same as Geogebra


Maple 2017.2 output:

 

dsolve({B(t)*(diff(B(t), t, t))*A(t)-A(t)*(diff(B(t), t))^2-(diff(A(t), t, t))*B(t)^2+(diff(A(t), t))*B(t)*(diff(B(t), t))-A(t) = 0, diff(A(t), t) = 0});

#[{B(t) = B(t)}, {A(t) = 0}], [{A(t) = _C3}, {B(t) = (1/2)*_C1*(1/((exp(_C2/_C1))^2*(exp(t/_C1))^2)+1)*exp(_C2/_C1)*exp(t/_C1), B(t) = (1/2)*_C1*((exp(_C2/_C1))^2*(exp(t/_C1))^2+1)/(exp(_C2/_C1)*exp(t/_C1))}]

 

Digits := 20;
sol := 126*0.9 = int(14*t*exp(-(1/3)*t), t = 0 .. x);
solve({sol, x > 0}, {x})

#{x = 11.669160509602287174}

As procedure:

UpperLimit := proc (percent) rhs(solve({(126/100)*percent = int(14*t*exp(-(1/3)*t), t = 0 .. x), 0 < x}, x)[1]) end proc;
UpperLimit(90)

#-3-3*LambertW(-1, -(1/10)*exp(-1))

evalf(UpperLimit(90))# 90%

# 11.669160509602287174

evalf(UpperLimit(50))# 50%

#5.0350409700499819602

evalf(UpperLimit(10))# 10%

#1.5954348251688360603

 


 

Sum(Sum((t+1)*Q[h]^2*(1-Q[h])^(t+T)/(t+1+(R[h]/S[h])^sigma*(T+1)), t = 0 .. infinity), T = 0 .. infinity)

Sum(Sum((t+1)*Q[h]^2*(1-Q[h])^(t+T)/(t+1+(R[h]/S[h])^sigma*(T+1)), t = 0 .. infinity), T = 0 .. infinity)

(1)

NULL

func := (t+1)*Q[h]^2*(1-Q[h])^(t+T)/(t+1+(R[h]/S[h])^sigma*(T+1))

(t+1)*Q[h]^2*(1-Q[h])^(t+T)/(t+1+(R[h]/S[h])^sigma*(T+1))

(2)

s1 := `assuming`([sum(func, t = 0 .. infinity, formal)], [Q[h] > 0, R[h] > 0, S[h] > 0, sigma > 0])

Q[h]^2*(1-Q[h])^T*((R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*S[h]^(-sigma)/Q[h]-R[h]^sigma*(R[h]^sigma*T^2+S[h]^sigma*T+2*R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*(S[h]^(-sigma))^2*LerchPhi(1-Q[h], 1, (R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*S[h]^(-sigma)))/((R[h]/S[h])^sigma*T+(R[h]/S[h])^sigma+1)

(3)

evalf(Sum(s1, T = 0 .. infinity))

Sum(Q[h]^2*(1-Q[h])^T*((R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*S[h]^(-sigma)/Q[h]-R[h]^sigma*(R[h]^sigma*T^2+S[h]^sigma*T+2*R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*(S[h]^(-sigma))^2*LerchPhi(1-Q[h], 1, (R[h]^sigma*T+S[h]^sigma+R[h]^sigma)*S[h]^(-sigma)))/((R[h]/S[h])^sigma*T+(R[h]/S[h])^sigma+1), T = 0 .. infinity)

(4)

s2 := `assuming`([sum(func, T = 0 .. infinity, formal)], [Q[h] > 0, R[h] > 0, S[h] > 0, sigma > 0])

(t+1)*Q[h]^2*(1-Q[h])^t*(t*S[h]^sigma+S[h]^sigma+R[h]^sigma)*R[h]^(-sigma)*LerchPhi(1-Q[h], 1, (t*S[h]^sigma+S[h]^sigma+R[h]^sigma)*R[h]^(-sigma))/(t+1+(R[h]/S[h])^sigma)

(5)

evalf(Sum(s2, t = 0 .. infinity))

Sum((t+1)*Q[h]^2*(1-Q[h])^t*(t*S[h]^sigma+S[h]^sigma+R[h]^sigma)*R[h]^(-sigma)*LerchPhi(1-Q[h], 1, (t*S[h]^sigma+S[h]^sigma+R[h]^sigma)*R[h]^(-sigma))/(t+1+(R[h]/S[h])^sigma), t = 0 .. infinity)

(6)

NULL


 

Download Sums.mw

Using SeriesCoefficient function:

Well, Maples convert(func,FPS) or convert(func,Sum) function is not strong enough.

Using Maxima http://maxima.sourceforge.net/  powerseries function:

tanh(x) = Sum((4^n-1)*4^n*bernoulli(2*n)*x^(2*n-1)/factorial(2*n), n = 0 .. infinity)

tanh(x+1) = Sum((4^n-1)*4^n*bernoulli(2*n)*(x+1)^(2*n-1)/factorial(2*n), n = 0 .. infinity)

 

Regards Mariusz

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