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Question: solution of differential equations

Question:solution of differential equations

emersondiaz 55 Maple 2023
ode syntax dsolve numeric differential-equations
October 22 2023
1

Good day to all the members who read this question.
I ask for your help to find the right way to solve this two differential equations (is attached the file). Used the common code "dsolve" but is not working on this problem. Doesn't matter if the solution is numeric or algebraic.
I will thaks a lot your kind help.
Best regards to all of you

DIFFERENTIAL_EQUATION.mw

restart

with(PDEtools)

First Part

The differential equation to solve:

(Delta*LinearAlgebra:-Transpose(D[1/2])*D__0-I*`μ__e`*D__0/(lambda+I*`μ__e`*r)-2*(`μ__e`^2*r^2+lambda^2))*R(r)

Definitions

D__n = `∂__r`+I*omega(a^2+r^2)/Delta+I*a*m/Delta+2*n*(r-M)/Delta

LinearAlgebra:-Transpose(D[n]) = `∂__r`-I*omega(a^2+r^2)/Delta-I*a*m/Delta+2*n*(r-M)/Delta

a := 1; M := 1; omega := 1; m := 1; `μ__e` := 1; lambda := 1

Delta := -2*M*r+a^2+r^2

I divided the differential equation in 3 parts (A, B, C).

A := (-I*omega(a^2+r^2)-I*a*m+r-M)*(diff(R(r), r)+I*omega(a^2+r^2)*R(r)/Delta+I*a*m*R(r)/Delta)+Delta*(diff(R(r), r, r))+I*omega(a^2+r^2)*(diff(R(r), r))+I*a*m*(diff(R(r), r))

B := -I*`μ__e`*(diff(R(r), r)+I*omega(a^2+r^2)*R(r)/Delta+I*a*m*R(r)/Delta)/(lambda+I*`μ__e`*r)

C := -(2*(`μ__e`^2*r^2+lambda^2))*R(r)

DE := A+B+C

E := dsolve(DE)

R(r) = DESol({diff(diff(_Y(r), r), r)-(1-r+I/(1+I*r))*(diff(_Y(r), r))/(r^2-2*r+1)-(-(2*I)*((-1-2*I)+r)/(r^2-2*r+1)-2/((1+I*r)*(r^2-2*r+1))+2*r^2+2)*_Y(r)/(r^2-2*r+1)}, {_Y(r)})

 (1)

dsolve({DE, DE(0) = 1}, numeric, range = 0 .. 20)

Error, (in dsolve/numeric/type_check) insufficient initial/boundary value information for procedure defined problem

  

Second Part

The differential equation to solve:

[`#msub(mi("L",fontweight = "bold"),mfrac(mn("1",fontweight = "bold"),mn("2",fontweight = "bold"),linethickness = "1"))`*LinearAlgebra:-Transpose(L[1/2])+a*`μ__e`*sin(theta)*LinearAlgebra:-Transpose(L[1/2])/(lambda+a*`μ__e`*cos(theta))+2*(lambda^2+a^2*`μ__e`*cos(theta)^2)]*S(theta) = 0

Definitions:

L__n = a*omega*`sinθ`+m*`cosecθ`+n*`cotθ`+`∂__θ`

LinearAlgebra:-Transpose(L[n]) = -a*omega*`sinθ`-m*`cosecθ`+n*`cotθ`+`∂__θ`

Also I divided the differential equation in 3 parts (A, B, C).

F := (omega*a*sin(theta)+m/sin(theta)+1/(2*tan(theta)))(diff(S(theta), theta)-omega*a*sin(theta)*S(theta)-m*S(theta)/sin(theta)+S(theta)/(2*tan(theta)))+diff(S(theta), theta, theta)-omega*a*sin(theta)*(diff(S(theta), theta))-m*(diff(S(theta), theta))/sin(theta)+(diff(S(theta), theta))/(2*tan(theta))

G := a*`μ__e`*sin(theta)*(diff(S(theta), theta)-omega*a*sin(theta)*S(theta)-m*S(theta)/sin(theta)+S(theta)/(2*tan(theta)))/(lambda+a*`μ__e`*cos(theta))

H := (2*(lambda^2+a^2*`μ__e`*cos(theta)^2))*S(theta)

DF := F+G+H

dsolve(DF)

dsolve({DF, DF(0) = 1}, numeric, range = 0 .. 20)

Error, (in dsolve/numeric/type_check) insufficient initial/boundary value information for procedure defined problem

  

NULL

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