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These are replies submitted by pooyan1990

@Carl Love 

Thanks. it worked.


Thank you so much. This was exactly what I need...


at first, it was immediately (2 second) after I press the enter button.

but when I impose assume to the entire code, it took several seconds (about 20).

@Carl Love 

several seconds about 20

@Rouben Rostamian  

thank you

B,H,L__1 are independent parameters and they are not relating to any parameter and I will give them different values for my work. they are actually my entrance parameters for my problem and the only way for me is to solve the ODE for symbolic B,H,L__1.

@Carl Love 

thank you for your answer

I did exactly what you said and I get the same error again!! "Error, (in dsolve) give the main variable as a second argument"

is it possible that the problem is with my maple?


@Carl Love 

thank you so much, worked for me


you can see the initial condition from the paper.



The initial condition from the paper for this differential equation is :


but as I stated in my first post, I can't implement this initial condition to this DE.

The error is like that:
Error, (in dsolve) unexpected occurence of the variables {beta,theta} in the forst operand... 
and you can see that completely in my first post

Is there anyway to implement this initial condition to the equation?

thank you


I have tried many times and again I have done it in your way and doesn't obtain the correct answer.

when solving the differential equation, an integral comes up and doesn't solve.

I have attached the worksheet.

would you please take a look at this?

eq48a is from the paper and sol2 should be equal to eq48a.

the paper's link in removed because of copyright violaton. if you need the paper let me know to send it to you.

thank you for taking the time


d := 10; c := 10; L__1 := 4; R__i := (1/2)*d+L__1; R__f := c+(1/2)*d+L__1;
eq43:= v__beta(r,theta,beta)= v__m(beta)*(1-(r^2/(R__max(theta,beta)^2)));
eq37:=diff(v__r(r, theta, beta),r)*r*(R__f-r*cos(theta))+v__r(r,theta,beta)*(R__f-2*r*cos(theta))+r*diff(v__beta(r,theta,beta),beta)=0;

d := 10

c := 10

L__1 := 4

R__i := 9

R__f := 19

eq43 := v__beta(r, theta, beta) = v__m(beta)*(1-r^2/R__max(theta, beta)^2)

eq37 := (diff(v__r(r, theta, beta), r))*r*(19-r*cos(theta))+v__r(r, theta, beta)*(19-2*r*cos(theta))+r*(diff(v__beta(r, theta, beta), beta)) = 0



eq39 := R(beta) = 9+20*beta/Pi


eq41:=R__max(theta,0)=((2*L__1*cos(theta)) + sqrt(d^2 - 2*(L__1)^2 + 2*(L__1)^2*cos(2*theta)))/2;

eq38 := R__max(theta, beta) = (1/9)*R__max(theta, 0)*R(beta)

eq41 := R__max(theta, 0) = 4*cos(theta)+(17+8*cos(2*theta))^(1/2)


eq38a:= subs(eq41, eq38);

eq38a := R__max(theta, beta) = (1/9)*(4*cos(theta)+(17+8*cos(2*theta))^(1/2))*R(beta)



eq38b := R__max(theta, beta) = (1/9)*(9*Pi+20*beta)*(4*cos(theta)+(17+8*cos(2*theta))^(1/2))/Pi


eq42:=R__max(theta,beta)=( (((d+2*L__1)*Pi) - 2*beta*(d+2*L__1-2*R__f))*(((2*L__1*cos(theta)) + sqrt(d^2 - 2*(L__1)^2 + 2*(L__1)^2*cos(2*theta)))))/(2*Pi*(d+2*L__1));

eq42 := R__max(theta, beta) = (1/36)*(18*pi+40*beta)*(8*cos(theta)+2*(17+8*cos(2*theta))^(1/2))/pi


eq44 := v__m(beta) = 1/(int(int((1-r^2/R__max(theta, beta)^2)*r, r = 0 .. R__max(theta, beta)), theta = 0 .. 2*Pi))

eq44 := v__m(beta) = 4/(int(R__max(theta, beta)^2, theta = 0 .. 2*Pi))



eq44a := v__m(beta) = (162/25)*Pi/(9*Pi+20*beta)^2



eq47 := v__beta(r, theta, beta) = (162/25)*Pi*(1-81*r^2*Pi^2/((9*Pi+20*beta)^2*(4*cos(theta)+(17+8*cos(2*theta))^(1/2))^2))/(9*Pi+20*beta)^2


eq37a:=simplify(expand(subs(eq47, eq37)));

eq37a := (1/5)*(-2361960*(r*cos(theta)-19)*(cos(theta)*(9+16*cos(theta)^2)^(1/2)+4*cos(theta)^2+9/8)*r*(Pi+(20/9)*beta)^5*(diff(v__r(r, theta, beta), r))-4723920*(r*cos(theta)-19/2)*(cos(theta)*(9+16*cos(theta)^2)^(1/2)+4*cos(theta)^2+9/8)*(Pi+(20/9)*beta)^5*v__r(r, theta, beta)-839808*(4*(Pi+(20/9)*beta)^2*cos(theta)^2+(9+16*cos(theta)^2)^(1/2)*(Pi+(20/9)*beta)^2*cos(theta)+(-(1/4)*r^2+9/8)*Pi^2+5*Pi*beta+(50/9)*beta^2)*Pi*r)/((9*Pi+20*beta)^5*(32*cos(theta)^2+8*cos(theta)*(9+16*cos(theta)^2)^(1/2)+9)) = 0


sol1:=simplify(dsolve(eq37a, v__r(r, theta,beta)));

sol1 := v__r(r, theta, beta) = (1/5)*(-1296*Pi*(Int(-162*(-2*(cos(2*theta)*r+r-38*cos(theta))*(Pi+(20/9)*beta)^2*(17+8*cos(2*theta))^(1/2)+152*(Pi+(20/9)*beta)^2*cos(2*theta)-4*r*(Pi+(20/9)*beta)^2*cos(3*theta)+(Pi^2*r^2-(33/2)*(Pi+(20/9)*beta)^2)*r*cos(theta)-19*Pi^2*r^2+(475/2)*(Pi+(20/9)*beta)^2)*r/((4*cos(2*theta)*r-152*cos(theta)+4*r)*(17+8*cos(2*theta))^(1/2)+33*r*cos(theta)+8*r*cos(3*theta)-304*cos(2*theta)-475), r))+295245*_F1(theta, beta)*(Pi+(20/9)*beta)^5)/((9*Pi+20*beta)^5*(r*cos(theta)-19)*r)


sol2:=eval(sol1, _F1(theta, beta)=0);

sol2 := v__r(r, theta, beta) = -(1296/5)*Pi*(Int(-162*(-2*(cos(2*theta)*r+r-38*cos(theta))*(Pi+(20/9)*beta)^2*(17+8*cos(2*theta))^(1/2)+152*(Pi+(20/9)*beta)^2*cos(2*theta)-4*r*(Pi+(20/9)*beta)^2*cos(3*theta)+(Pi^2*r^2-(33/2)*(Pi+(20/9)*beta)^2)*r*cos(theta)-19*Pi^2*r^2+(475/2)*(Pi+(20/9)*beta)^2)*r/((4*cos(2*theta)*r-152*cos(theta)+4*r)*(17+8*cos(2*theta))^(1/2)+33*r*cos(theta)+8*r*cos(3*theta)-304*cos(2*theta)-475), r))/((9*Pi+20*beta)^5*(r*cos(theta)-19)*r)


eq0 := o__1 = (1-2*L__1/d)^2*(1+2*L__1/d+4*c*beta/(d*Pi))^2; eq00 := o__2 = 1-4*L__1*(cos(theta)*sqrt(1-4*L__1^2*sin(theta)^2/d^2)-L__1*cos(theta)/d)/d

eq0 := o__1 = (1/25)*(9/5+4*beta/Pi)^2

eq00 := o__2 = 1-(8/25)*cos(theta)*(25-16*sin(theta)^2)^(1/2)+(16/25)*cos(theta)


eq48 := v__r(r, theta, beta) = 8*(1+2*L__1/d)^2*c*r*(o__1-4*(r/d)^2*o__2)/(d^2*Pi*(1-2*L__1/d)^2*(1+2*L__1/d+4*c*beta/(d*Pi))^5*(R__f/d-r*cos(theta)/d))

eq48 := v__r(r, theta, beta) = (324/5)*r*(o__1-(1/25)*r^2*o__2)/(Pi*(9/5+4*beta/Pi)^5*(19/10-(1/10)*r*cos(theta)))


eq48a := simplify(expand(subs([eq0, eq00], eq48)))

eq48a := v__r(r, theta, beta) = -25920*Pi^2*(Pi^2*(9+16*cos(theta)^2)^(1/2)*cos(theta)*r^2-2*Pi^2*cos(theta)*r^2+(-(25/8)*r^2+81/8)*Pi^2+45*Pi*beta+50*beta^2)*r/((9*Pi+20*beta)^5*(r*cos(theta)-19))





Thank you for taking the time

but there some mistakes in this solution.

In this paper there two different mechanism and equations and you use equations for the first one but I want the second one.

V(beta) is not eq1 in the paper, it is eq.43

Our final answer for V(r) is not eq.2, it is eq.48

we also have an initial condition in page 10 of the paper which I highlighted that above eq.48.

I substitute these equations and didn't get the right answer.

would you please do this for me with these reforms?

I will be grateful for your help.


Thanks for your answer

I Understand this but I can't change my variable names. I extracted these formulations from a valid paper so I think they are correct. I have uploaded the relevant article pdf. you can see all of these equations in pages 9 and 10 from eq. 37 to 48. the boundry condition is also highlighted in page 10. I would be grateful if you take a brief look at this article and if you have any idea to solve this problem.

thanks again

[Edit: .pdf deleted because of copyright violation.]

@Christian Wolinski 

Thanks for your answer

I have solved the problem. I just forgot to Multiply Beta with other terms and there was no sign of multiplying between them. sorry for the inconvenience.

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