Solving second order ordinary differential equation with variable constants











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I'm having trouble solving a differential equation I found:



$$ u''(x) + xint_0^xu(t)dt = f(x) $$



where: $ xin[0,1], quad u(0) = 1, quad u(1) = -1 $, and $f(x)$ any given function.



One of my problems is I don't really understand the term of the integral because with that it doesn't look like a linear second order ordinary differential equation or anything I have ever seen before. What I have tried is converting it to a system of differential equations and also tried to solve it as a homogenous equation, but it didn't turned out into anything useful,



If anyone can help me getting started I would be very grateful.










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  • 1




    Let $v(x) = int_0^x u(t) , mathrm dt$. Then $v'(x) = u(x)$, and your equation becomes $$ v'''(x) +xv(x) = f(x). $$
    – MisterRiemann
    Nov 23 at 17:57















up vote
0
down vote

favorite












I'm having trouble solving a differential equation I found:



$$ u''(x) + xint_0^xu(t)dt = f(x) $$



where: $ xin[0,1], quad u(0) = 1, quad u(1) = -1 $, and $f(x)$ any given function.



One of my problems is I don't really understand the term of the integral because with that it doesn't look like a linear second order ordinary differential equation or anything I have ever seen before. What I have tried is converting it to a system of differential equations and also tried to solve it as a homogenous equation, but it didn't turned out into anything useful,



If anyone can help me getting started I would be very grateful.










share|cite|improve this question




















  • 1




    Let $v(x) = int_0^x u(t) , mathrm dt$. Then $v'(x) = u(x)$, and your equation becomes $$ v'''(x) +xv(x) = f(x). $$
    – MisterRiemann
    Nov 23 at 17:57













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I'm having trouble solving a differential equation I found:



$$ u''(x) + xint_0^xu(t)dt = f(x) $$



where: $ xin[0,1], quad u(0) = 1, quad u(1) = -1 $, and $f(x)$ any given function.



One of my problems is I don't really understand the term of the integral because with that it doesn't look like a linear second order ordinary differential equation or anything I have ever seen before. What I have tried is converting it to a system of differential equations and also tried to solve it as a homogenous equation, but it didn't turned out into anything useful,



If anyone can help me getting started I would be very grateful.










share|cite|improve this question















I'm having trouble solving a differential equation I found:



$$ u''(x) + xint_0^xu(t)dt = f(x) $$



where: $ xin[0,1], quad u(0) = 1, quad u(1) = -1 $, and $f(x)$ any given function.



One of my problems is I don't really understand the term of the integral because with that it doesn't look like a linear second order ordinary differential equation or anything I have ever seen before. What I have tried is converting it to a system of differential equations and also tried to solve it as a homogenous equation, but it didn't turned out into anything useful,



If anyone can help me getting started I would be very grateful.







differential-equations computational-mathematics integral-equations






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edited Nov 24 at 12:40









doraemonpaul

12.5k31660




12.5k31660










asked Nov 23 at 17:34









hugdelcur96

165




165








  • 1




    Let $v(x) = int_0^x u(t) , mathrm dt$. Then $v'(x) = u(x)$, and your equation becomes $$ v'''(x) +xv(x) = f(x). $$
    – MisterRiemann
    Nov 23 at 17:57














  • 1




    Let $v(x) = int_0^x u(t) , mathrm dt$. Then $v'(x) = u(x)$, and your equation becomes $$ v'''(x) +xv(x) = f(x). $$
    – MisterRiemann
    Nov 23 at 17:57








1




1




Let $v(x) = int_0^x u(t) , mathrm dt$. Then $v'(x) = u(x)$, and your equation becomes $$ v'''(x) +xv(x) = f(x). $$
– MisterRiemann
Nov 23 at 17:57




Let $v(x) = int_0^x u(t) , mathrm dt$. Then $v'(x) = u(x)$, and your equation becomes $$ v'''(x) +xv(x) = f(x). $$
– MisterRiemann
Nov 23 at 17:57










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With $U(x)=int_0^x u(t),dt$ you get $U'=u$, $U(0)=0$ and inserted it gives a third order ODE
$$
U'''(x)+xU(x)=f(x)
$$

with initial conditions $U(0)=0$, $U'(0)=u(0)=1$ and $U'(1)=u(1)=-1$.






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    down vote













    With $U(x)=int_0^x u(t),dt$ you get $U'=u$, $U(0)=0$ and inserted it gives a third order ODE
    $$
    U'''(x)+xU(x)=f(x)
    $$

    with initial conditions $U(0)=0$, $U'(0)=u(0)=1$ and $U'(1)=u(1)=-1$.






    share|cite|improve this answer

























      up vote
      2
      down vote













      With $U(x)=int_0^x u(t),dt$ you get $U'=u$, $U(0)=0$ and inserted it gives a third order ODE
      $$
      U'''(x)+xU(x)=f(x)
      $$

      with initial conditions $U(0)=0$, $U'(0)=u(0)=1$ and $U'(1)=u(1)=-1$.






      share|cite|improve this answer























        up vote
        2
        down vote










        up vote
        2
        down vote









        With $U(x)=int_0^x u(t),dt$ you get $U'=u$, $U(0)=0$ and inserted it gives a third order ODE
        $$
        U'''(x)+xU(x)=f(x)
        $$

        with initial conditions $U(0)=0$, $U'(0)=u(0)=1$ and $U'(1)=u(1)=-1$.






        share|cite|improve this answer












        With $U(x)=int_0^x u(t),dt$ you get $U'=u$, $U(0)=0$ and inserted it gives a third order ODE
        $$
        U'''(x)+xU(x)=f(x)
        $$

        with initial conditions $U(0)=0$, $U'(0)=u(0)=1$ and $U'(1)=u(1)=-1$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 23 at 17:57









        LutzL

        55.3k42053




        55.3k42053






























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