How to obtain and work with steady-state part of DE solution?

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I'm trying to obtain resonance curves of the system. System can be described as



F,m,k:=2,1,4: 
lambda:= beta/(2*m):
omega:=sqrt(k/m):
de:=diff(x(t),t$2)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
cond:=x(0)=0, D(x)(0)=0:
sol := dsolve({cond, de});


Solving gives sum of terms, some of which "die out" with time (since these terms have exp(-...*t)) and some of which form steady-state solution (solution for t -> ∞). This solution will be in form xstst=f(gamma1)*sin(...). In order to obtain resonance curves, I need to plot f(gamma1) (for chosen constant betas, say, 2,1,0.5,0.25,etc.).



I've solved this "by hand" and found f := F/(sqrt((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)). Plotting this for any chosen beta gives the result needed, for example, for beta:=0.5 the plot is enter image description here



I wonder if I can obtain these curves using maple functions only (without solving anything "by hand" at all).



[edited]enter image description here










share|improve this question































    0















    I'm trying to obtain resonance curves of the system. System can be described as



    F,m,k:=2,1,4: 
    lambda:= beta/(2*m):
    omega:=sqrt(k/m):
    de:=diff(x(t),t$2)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
    cond:=x(0)=0, D(x)(0)=0:
    sol := dsolve({cond, de});


    Solving gives sum of terms, some of which "die out" with time (since these terms have exp(-...*t)) and some of which form steady-state solution (solution for t -> ∞). This solution will be in form xstst=f(gamma1)*sin(...). In order to obtain resonance curves, I need to plot f(gamma1) (for chosen constant betas, say, 2,1,0.5,0.25,etc.).



    I've solved this "by hand" and found f := F/(sqrt((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)). Plotting this for any chosen beta gives the result needed, for example, for beta:=0.5 the plot is enter image description here



    I wonder if I can obtain these curves using maple functions only (without solving anything "by hand" at all).



    [edited]enter image description here










    share|improve this question



























      0












      0








      0








      I'm trying to obtain resonance curves of the system. System can be described as



      F,m,k:=2,1,4: 
      lambda:= beta/(2*m):
      omega:=sqrt(k/m):
      de:=diff(x(t),t$2)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
      cond:=x(0)=0, D(x)(0)=0:
      sol := dsolve({cond, de});


      Solving gives sum of terms, some of which "die out" with time (since these terms have exp(-...*t)) and some of which form steady-state solution (solution for t -> ∞). This solution will be in form xstst=f(gamma1)*sin(...). In order to obtain resonance curves, I need to plot f(gamma1) (for chosen constant betas, say, 2,1,0.5,0.25,etc.).



      I've solved this "by hand" and found f := F/(sqrt((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)). Plotting this for any chosen beta gives the result needed, for example, for beta:=0.5 the plot is enter image description here



      I wonder if I can obtain these curves using maple functions only (without solving anything "by hand" at all).



      [edited]enter image description here










      share|improve this question
















      I'm trying to obtain resonance curves of the system. System can be described as



      F,m,k:=2,1,4: 
      lambda:= beta/(2*m):
      omega:=sqrt(k/m):
      de:=diff(x(t),t$2)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
      cond:=x(0)=0, D(x)(0)=0:
      sol := dsolve({cond, de});


      Solving gives sum of terms, some of which "die out" with time (since these terms have exp(-...*t)) and some of which form steady-state solution (solution for t -> ∞). This solution will be in form xstst=f(gamma1)*sin(...). In order to obtain resonance curves, I need to plot f(gamma1) (for chosen constant betas, say, 2,1,0.5,0.25,etc.).



      I've solved this "by hand" and found f := F/(sqrt((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)). Plotting this for any chosen beta gives the result needed, for example, for beta:=0.5 the plot is enter image description here



      I wonder if I can obtain these curves using maple functions only (without solving anything "by hand" at all).



      [edited]enter image description here







      maple






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited Jan 17 at 13:39







      Kelly Shepphard

















      asked Jan 3 at 21:38









      Kelly ShepphardKelly Shepphard

      11616




      11616
























          2 Answers
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          1














          It does not make sense to expect Maple to represent the result in terms of either sin(theta) or cos(theta), using some formulas for those terms that appear nowhere in the problem specification and are entirely introduced by you.



          The following is obtained using a radical (square root) in each of cond1 and cond2.



          restart;
          de := diff(x(t),t,t)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
          cond := x(0)=0, D(x)(0)=0:

          sol := dsolve({cond, de}):

          E,T := selectremove(hastype,rhs(sol),specfunc(anything,exp)):

          lprint(T);

          F*(2*sin(gamma1*t)*gamma1*lambda-cos(gamma1*t)*gamma1^2
          +cos(gamma1*t)*omega^2)/(gamma1^4+4*gamma1^2*lambda^2-
          2*gamma1^2*omega^2+omega^4)

          cond1 := cos(theta) = (-gamma1^2+omega^2)
          /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

          cond2 := sin(theta) = (-2*lambda*gamma1)
          /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

          frontend(algsubs, [numer(rhs(cond1))=lhs(cond1)*denom(rhs(cond1)),
          numer(T)],
          [{`+`,`*`,`=`},{}]):

          frontend(algsubs, [numer(rhs(cond2))=lhs(cond2)*denom(rhs(cond2)),
          %],
          [{`+`,`*`,`=`},{}]):

          ans := collect(combine(%, trig),cos)/denom(T):

          lprint(ans);

          F*cos(gamma1*t+theta)/(gamma1^4+4*gamma1^2*lambda^2-
          2*gamma1^2*omega^2+omega^4)^(1/2)

          subsans := eval(eval(ans,[lambda=beta/(2*m),omega=sqrt(k/m)]),
          [F=2,m=1,k=4]):

          lprint(subsans);

          2*cos(gamma1*t+theta)
          /(beta^2*gamma1^2+gamma1^4-8*gamma1^2+16)^(1/2)





          share|improve this answer































            1














            I'm not sure I fully understand your question, but when I run your code I get some terms with exponentials, some with sines and some with cosines. You can grab the coefficient of the sine terms with



            coeff( collect( rhs( sol ) , sin( gamma1 * t ) ) , sin( gamma1 * t ) , 1 )





            share|improve this answer
























            • Thank you for the answer, but I need to find coeff only for the last summand (which doesn't contain exp(-t*...)) and term will be like f(gamma1)*sin(gamma1*t+theta), where theta is also a function of lambda,omega and gamma1. I've tried to collect this, but it seems that it won't work (collecting gives the same 3rd term as before). At least, now I know that it's impossible. +1 for the help, thank you. This idea was very useful for understanding.

              – Kelly Shepphard
              Jan 17 at 6:43











            • @KellyShepphard --- I suspect it is possible, but I'm still not 100% sure of exactly what you want. (There is no theta in your code, and I don't know where the sqrt comes from in the work you did by hand.) Perhaps I can give a better answer if you upload a screenshot from Maple and highlight the terms that you want to isolate.

              – Ian Thompson
              Jan 17 at 9:19











            • I've added screenshot. Sorry for the lack of info, I've thought Maple automatically solves it as was needed.

              – Kelly Shepphard
              Jan 17 at 13:40












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            2 Answers
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            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1














            It does not make sense to expect Maple to represent the result in terms of either sin(theta) or cos(theta), using some formulas for those terms that appear nowhere in the problem specification and are entirely introduced by you.



            The following is obtained using a radical (square root) in each of cond1 and cond2.



            restart;
            de := diff(x(t),t,t)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
            cond := x(0)=0, D(x)(0)=0:

            sol := dsolve({cond, de}):

            E,T := selectremove(hastype,rhs(sol),specfunc(anything,exp)):

            lprint(T);

            F*(2*sin(gamma1*t)*gamma1*lambda-cos(gamma1*t)*gamma1^2
            +cos(gamma1*t)*omega^2)/(gamma1^4+4*gamma1^2*lambda^2-
            2*gamma1^2*omega^2+omega^4)

            cond1 := cos(theta) = (-gamma1^2+omega^2)
            /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

            cond2 := sin(theta) = (-2*lambda*gamma1)
            /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

            frontend(algsubs, [numer(rhs(cond1))=lhs(cond1)*denom(rhs(cond1)),
            numer(T)],
            [{`+`,`*`,`=`},{}]):

            frontend(algsubs, [numer(rhs(cond2))=lhs(cond2)*denom(rhs(cond2)),
            %],
            [{`+`,`*`,`=`},{}]):

            ans := collect(combine(%, trig),cos)/denom(T):

            lprint(ans);

            F*cos(gamma1*t+theta)/(gamma1^4+4*gamma1^2*lambda^2-
            2*gamma1^2*omega^2+omega^4)^(1/2)

            subsans := eval(eval(ans,[lambda=beta/(2*m),omega=sqrt(k/m)]),
            [F=2,m=1,k=4]):

            lprint(subsans);

            2*cos(gamma1*t+theta)
            /(beta^2*gamma1^2+gamma1^4-8*gamma1^2+16)^(1/2)





            share|improve this answer




























              1














              It does not make sense to expect Maple to represent the result in terms of either sin(theta) or cos(theta), using some formulas for those terms that appear nowhere in the problem specification and are entirely introduced by you.



              The following is obtained using a radical (square root) in each of cond1 and cond2.



              restart;
              de := diff(x(t),t,t)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
              cond := x(0)=0, D(x)(0)=0:

              sol := dsolve({cond, de}):

              E,T := selectremove(hastype,rhs(sol),specfunc(anything,exp)):

              lprint(T);

              F*(2*sin(gamma1*t)*gamma1*lambda-cos(gamma1*t)*gamma1^2
              +cos(gamma1*t)*omega^2)/(gamma1^4+4*gamma1^2*lambda^2-
              2*gamma1^2*omega^2+omega^4)

              cond1 := cos(theta) = (-gamma1^2+omega^2)
              /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

              cond2 := sin(theta) = (-2*lambda*gamma1)
              /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

              frontend(algsubs, [numer(rhs(cond1))=lhs(cond1)*denom(rhs(cond1)),
              numer(T)],
              [{`+`,`*`,`=`},{}]):

              frontend(algsubs, [numer(rhs(cond2))=lhs(cond2)*denom(rhs(cond2)),
              %],
              [{`+`,`*`,`=`},{}]):

              ans := collect(combine(%, trig),cos)/denom(T):

              lprint(ans);

              F*cos(gamma1*t+theta)/(gamma1^4+4*gamma1^2*lambda^2-
              2*gamma1^2*omega^2+omega^4)^(1/2)

              subsans := eval(eval(ans,[lambda=beta/(2*m),omega=sqrt(k/m)]),
              [F=2,m=1,k=4]):

              lprint(subsans);

              2*cos(gamma1*t+theta)
              /(beta^2*gamma1^2+gamma1^4-8*gamma1^2+16)^(1/2)





              share|improve this answer


























                1












                1








                1







                It does not make sense to expect Maple to represent the result in terms of either sin(theta) or cos(theta), using some formulas for those terms that appear nowhere in the problem specification and are entirely introduced by you.



                The following is obtained using a radical (square root) in each of cond1 and cond2.



                restart;
                de := diff(x(t),t,t)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
                cond := x(0)=0, D(x)(0)=0:

                sol := dsolve({cond, de}):

                E,T := selectremove(hastype,rhs(sol),specfunc(anything,exp)):

                lprint(T);

                F*(2*sin(gamma1*t)*gamma1*lambda-cos(gamma1*t)*gamma1^2
                +cos(gamma1*t)*omega^2)/(gamma1^4+4*gamma1^2*lambda^2-
                2*gamma1^2*omega^2+omega^4)

                cond1 := cos(theta) = (-gamma1^2+omega^2)
                /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

                cond2 := sin(theta) = (-2*lambda*gamma1)
                /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

                frontend(algsubs, [numer(rhs(cond1))=lhs(cond1)*denom(rhs(cond1)),
                numer(T)],
                [{`+`,`*`,`=`},{}]):

                frontend(algsubs, [numer(rhs(cond2))=lhs(cond2)*denom(rhs(cond2)),
                %],
                [{`+`,`*`,`=`},{}]):

                ans := collect(combine(%, trig),cos)/denom(T):

                lprint(ans);

                F*cos(gamma1*t+theta)/(gamma1^4+4*gamma1^2*lambda^2-
                2*gamma1^2*omega^2+omega^4)^(1/2)

                subsans := eval(eval(ans,[lambda=beta/(2*m),omega=sqrt(k/m)]),
                [F=2,m=1,k=4]):

                lprint(subsans);

                2*cos(gamma1*t+theta)
                /(beta^2*gamma1^2+gamma1^4-8*gamma1^2+16)^(1/2)





                share|improve this answer













                It does not make sense to expect Maple to represent the result in terms of either sin(theta) or cos(theta), using some formulas for those terms that appear nowhere in the problem specification and are entirely introduced by you.



                The following is obtained using a radical (square root) in each of cond1 and cond2.



                restart;
                de := diff(x(t),t,t)+2*lambda*diff(x(t),t)+omega^2*x(t)=F*cos(gamma1*t):
                cond := x(0)=0, D(x)(0)=0:

                sol := dsolve({cond, de}):

                E,T := selectremove(hastype,rhs(sol),specfunc(anything,exp)):

                lprint(T);

                F*(2*sin(gamma1*t)*gamma1*lambda-cos(gamma1*t)*gamma1^2
                +cos(gamma1*t)*omega^2)/(gamma1^4+4*gamma1^2*lambda^2-
                2*gamma1^2*omega^2+omega^4)

                cond1 := cos(theta) = (-gamma1^2+omega^2)
                /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

                cond2 := sin(theta) = (-2*lambda*gamma1)
                /((-gamma1^2+omega^2)^2+4*lambda^2*gamma1^2)^(1/2):

                frontend(algsubs, [numer(rhs(cond1))=lhs(cond1)*denom(rhs(cond1)),
                numer(T)],
                [{`+`,`*`,`=`},{}]):

                frontend(algsubs, [numer(rhs(cond2))=lhs(cond2)*denom(rhs(cond2)),
                %],
                [{`+`,`*`,`=`},{}]):

                ans := collect(combine(%, trig),cos)/denom(T):

                lprint(ans);

                F*cos(gamma1*t+theta)/(gamma1^4+4*gamma1^2*lambda^2-
                2*gamma1^2*omega^2+omega^4)^(1/2)

                subsans := eval(eval(ans,[lambda=beta/(2*m),omega=sqrt(k/m)]),
                [F=2,m=1,k=4]):

                lprint(subsans);

                2*cos(gamma1*t+theta)
                /(beta^2*gamma1^2+gamma1^4-8*gamma1^2+16)^(1/2)






                share|improve this answer












                share|improve this answer



                share|improve this answer










                answered Jan 31 at 10:12









                aceracer

                5,0071015




                5,0071015

























                    1














                    I'm not sure I fully understand your question, but when I run your code I get some terms with exponentials, some with sines and some with cosines. You can grab the coefficient of the sine terms with



                    coeff( collect( rhs( sol ) , sin( gamma1 * t ) ) , sin( gamma1 * t ) , 1 )





                    share|improve this answer
























                    • Thank you for the answer, but I need to find coeff only for the last summand (which doesn't contain exp(-t*...)) and term will be like f(gamma1)*sin(gamma1*t+theta), where theta is also a function of lambda,omega and gamma1. I've tried to collect this, but it seems that it won't work (collecting gives the same 3rd term as before). At least, now I know that it's impossible. +1 for the help, thank you. This idea was very useful for understanding.

                      – Kelly Shepphard
                      Jan 17 at 6:43











                    • @KellyShepphard --- I suspect it is possible, but I'm still not 100% sure of exactly what you want. (There is no theta in your code, and I don't know where the sqrt comes from in the work you did by hand.) Perhaps I can give a better answer if you upload a screenshot from Maple and highlight the terms that you want to isolate.

                      – Ian Thompson
                      Jan 17 at 9:19











                    • I've added screenshot. Sorry for the lack of info, I've thought Maple automatically solves it as was needed.

                      – Kelly Shepphard
                      Jan 17 at 13:40
















                    1














                    I'm not sure I fully understand your question, but when I run your code I get some terms with exponentials, some with sines and some with cosines. You can grab the coefficient of the sine terms with



                    coeff( collect( rhs( sol ) , sin( gamma1 * t ) ) , sin( gamma1 * t ) , 1 )





                    share|improve this answer
























                    • Thank you for the answer, but I need to find coeff only for the last summand (which doesn't contain exp(-t*...)) and term will be like f(gamma1)*sin(gamma1*t+theta), where theta is also a function of lambda,omega and gamma1. I've tried to collect this, but it seems that it won't work (collecting gives the same 3rd term as before). At least, now I know that it's impossible. +1 for the help, thank you. This idea was very useful for understanding.

                      – Kelly Shepphard
                      Jan 17 at 6:43











                    • @KellyShepphard --- I suspect it is possible, but I'm still not 100% sure of exactly what you want. (There is no theta in your code, and I don't know where the sqrt comes from in the work you did by hand.) Perhaps I can give a better answer if you upload a screenshot from Maple and highlight the terms that you want to isolate.

                      – Ian Thompson
                      Jan 17 at 9:19











                    • I've added screenshot. Sorry for the lack of info, I've thought Maple automatically solves it as was needed.

                      – Kelly Shepphard
                      Jan 17 at 13:40














                    1












                    1








                    1







                    I'm not sure I fully understand your question, but when I run your code I get some terms with exponentials, some with sines and some with cosines. You can grab the coefficient of the sine terms with



                    coeff( collect( rhs( sol ) , sin( gamma1 * t ) ) , sin( gamma1 * t ) , 1 )





                    share|improve this answer













                    I'm not sure I fully understand your question, but when I run your code I get some terms with exponentials, some with sines and some with cosines. You can grab the coefficient of the sine terms with



                    coeff( collect( rhs( sol ) , sin( gamma1 * t ) ) , sin( gamma1 * t ) , 1 )






                    share|improve this answer












                    share|improve this answer



                    share|improve this answer










                    answered Jan 16 at 22:34









                    Ian ThompsonIan Thompson

                    1464




                    1464













                    • Thank you for the answer, but I need to find coeff only for the last summand (which doesn't contain exp(-t*...)) and term will be like f(gamma1)*sin(gamma1*t+theta), where theta is also a function of lambda,omega and gamma1. I've tried to collect this, but it seems that it won't work (collecting gives the same 3rd term as before). At least, now I know that it's impossible. +1 for the help, thank you. This idea was very useful for understanding.

                      – Kelly Shepphard
                      Jan 17 at 6:43











                    • @KellyShepphard --- I suspect it is possible, but I'm still not 100% sure of exactly what you want. (There is no theta in your code, and I don't know where the sqrt comes from in the work you did by hand.) Perhaps I can give a better answer if you upload a screenshot from Maple and highlight the terms that you want to isolate.

                      – Ian Thompson
                      Jan 17 at 9:19











                    • I've added screenshot. Sorry for the lack of info, I've thought Maple automatically solves it as was needed.

                      – Kelly Shepphard
                      Jan 17 at 13:40



















                    • Thank you for the answer, but I need to find coeff only for the last summand (which doesn't contain exp(-t*...)) and term will be like f(gamma1)*sin(gamma1*t+theta), where theta is also a function of lambda,omega and gamma1. I've tried to collect this, but it seems that it won't work (collecting gives the same 3rd term as before). At least, now I know that it's impossible. +1 for the help, thank you. This idea was very useful for understanding.

                      – Kelly Shepphard
                      Jan 17 at 6:43











                    • @KellyShepphard --- I suspect it is possible, but I'm still not 100% sure of exactly what you want. (There is no theta in your code, and I don't know where the sqrt comes from in the work you did by hand.) Perhaps I can give a better answer if you upload a screenshot from Maple and highlight the terms that you want to isolate.

                      – Ian Thompson
                      Jan 17 at 9:19











                    • I've added screenshot. Sorry for the lack of info, I've thought Maple automatically solves it as was needed.

                      – Kelly Shepphard
                      Jan 17 at 13:40

















                    Thank you for the answer, but I need to find coeff only for the last summand (which doesn't contain exp(-t*...)) and term will be like f(gamma1)*sin(gamma1*t+theta), where theta is also a function of lambda,omega and gamma1. I've tried to collect this, but it seems that it won't work (collecting gives the same 3rd term as before). At least, now I know that it's impossible. +1 for the help, thank you. This idea was very useful for understanding.

                    – Kelly Shepphard
                    Jan 17 at 6:43





                    Thank you for the answer, but I need to find coeff only for the last summand (which doesn't contain exp(-t*...)) and term will be like f(gamma1)*sin(gamma1*t+theta), where theta is also a function of lambda,omega and gamma1. I've tried to collect this, but it seems that it won't work (collecting gives the same 3rd term as before). At least, now I know that it's impossible. +1 for the help, thank you. This idea was very useful for understanding.

                    – Kelly Shepphard
                    Jan 17 at 6:43













                    @KellyShepphard --- I suspect it is possible, but I'm still not 100% sure of exactly what you want. (There is no theta in your code, and I don't know where the sqrt comes from in the work you did by hand.) Perhaps I can give a better answer if you upload a screenshot from Maple and highlight the terms that you want to isolate.

                    – Ian Thompson
                    Jan 17 at 9:19





                    @KellyShepphard --- I suspect it is possible, but I'm still not 100% sure of exactly what you want. (There is no theta in your code, and I don't know where the sqrt comes from in the work you did by hand.) Perhaps I can give a better answer if you upload a screenshot from Maple and highlight the terms that you want to isolate.

                    – Ian Thompson
                    Jan 17 at 9:19













                    I've added screenshot. Sorry for the lack of info, I've thought Maple automatically solves it as was needed.

                    – Kelly Shepphard
                    Jan 17 at 13:40





                    I've added screenshot. Sorry for the lack of info, I've thought Maple automatically solves it as was needed.

                    – Kelly Shepphard
                    Jan 17 at 13:40


















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