How to prove the following identity regarding Laplace transforms?

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I tried solving it by integrating by parts but i was unsuccessful.




$$cal Lleft[int_0^xf(x-t)g(t) dtright]=F(p)G(p)$$







differential-equations laplace-transform







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  • 3




    Start from the right hand side.
    – Nosrati
    4 hours ago










  • @Nosrati i am getting two integrals multiplied together.how to bring them in single one ?
    – RockDock
    4 hours ago







  • 2




    by changing variables.
    – Nosrati
    4 hours ago














up vote
4
down vote

favorite
1












I tried solving it by integrating by parts but i was unsuccessful.




$$cal Lleft[int_0^xf(x-t)g(t) dtright]=F(p)G(p)$$







differential-equations laplace-transform







share|cite|improve this question



















  • 3




    Start from the right hand side.
    – Nosrati
    4 hours ago










  • @Nosrati i am getting two integrals multiplied together.how to bring them in single one ?
    – RockDock
    4 hours ago







  • 2




    by changing variables.
    – Nosrati
    4 hours ago












up vote
4
down vote

favorite
1









up vote
4
down vote

favorite
1






1





I tried solving it by integrating by parts but i was unsuccessful.




$$cal Lleft[int_0^xf(x-t)g(t) dtright]=F(p)G(p)$$







differential-equations laplace-transform







share|cite|improve this question















I tried solving it by integrating by parts but i was unsuccessful.




$$cal Lleft[int_0^xf(x-t)g(t) dtright]=F(p)G(p)$$







differential-equations laplace-transform




differential-equations laplace-transform






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 32 mins ago









Federico Poloni

2,4091325




2,4091325










asked 4 hours ago









RockDock

533




533







  • 3




    Start from the right hand side.
    – Nosrati
    4 hours ago










  • @Nosrati i am getting two integrals multiplied together.how to bring them in single one ?
    – RockDock
    4 hours ago







  • 2




    by changing variables.
    – Nosrati
    4 hours ago












  • 3




    Start from the right hand side.
    – Nosrati
    4 hours ago










  • @Nosrati i am getting two integrals multiplied together.how to bring them in single one ?
    – RockDock
    4 hours ago







  • 2




    by changing variables.
    – Nosrati
    4 hours ago







3




3




Start from the right hand side.
– Nosrati
4 hours ago




Start from the right hand side.
– Nosrati
4 hours ago












@Nosrati i am getting two integrals multiplied together.how to bring them in single one ?
– RockDock
4 hours ago





@Nosrati i am getting two integrals multiplied together.how to bring them in single one ?
– RockDock
4 hours ago





2




2




by changing variables.
– Nosrati
4 hours ago




by changing variables.
– Nosrati
4 hours ago










1 Answer
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up vote
6
down vote



accepted










beginalign
F(p) G(p)
&= int_0^infty e^-puf(u) duint_0^infty e^-pvg(v) dv \
&= int_0^inftyint_0^infty e^-p(u+v)f(u)g(v) du dv ,,, , ,,, textu+v=t ,, , ,, textv=x\
&= int_0^inftyint_x^infty e^-ptf(t-x)g(x) dt dx ,, , ,, textchanging order of integration\
&= int_0^infty e^-ptBig[int_0^tf(t-x)g(x) dx Big] dt \
&= cal L(f*g)(t)
endalign






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    1 Answer
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    up vote
    6
    down vote



    accepted










    beginalign
    F(p) G(p)
    &= int_0^infty e^-puf(u) duint_0^infty e^-pvg(v) dv \
    &= int_0^inftyint_0^infty e^-p(u+v)f(u)g(v) du dv ,,, , ,,, textu+v=t ,, , ,, textv=x\
    &= int_0^inftyint_x^infty e^-ptf(t-x)g(x) dt dx ,, , ,, textchanging order of integration\
    &= int_0^infty e^-ptBig[int_0^tf(t-x)g(x) dx Big] dt \
    &= cal L(f*g)(t)
    endalign






    share|cite|improve this answer


























      up vote
      6
      down vote



      accepted










      beginalign
      F(p) G(p)
      &= int_0^infty e^-puf(u) duint_0^infty e^-pvg(v) dv \
      &= int_0^inftyint_0^infty e^-p(u+v)f(u)g(v) du dv ,,, , ,,, textu+v=t ,, , ,, textv=x\
      &= int_0^inftyint_x^infty e^-ptf(t-x)g(x) dt dx ,, , ,, textchanging order of integration\
      &= int_0^infty e^-ptBig[int_0^tf(t-x)g(x) dx Big] dt \
      &= cal L(f*g)(t)
      endalign






      share|cite|improve this answer
























        up vote
        6
        down vote



        accepted







        up vote
        6
        down vote



        accepted






        beginalign
        F(p) G(p)
        &= int_0^infty e^-puf(u) duint_0^infty e^-pvg(v) dv \
        &= int_0^inftyint_0^infty e^-p(u+v)f(u)g(v) du dv ,,, , ,,, textu+v=t ,, , ,, textv=x\
        &= int_0^inftyint_x^infty e^-ptf(t-x)g(x) dt dx ,, , ,, textchanging order of integration\
        &= int_0^infty e^-ptBig[int_0^tf(t-x)g(x) dx Big] dt \
        &= cal L(f*g)(t)
        endalign






        share|cite|improve this answer














        beginalign
        F(p) G(p)
        &= int_0^infty e^-puf(u) duint_0^infty e^-pvg(v) dv \
        &= int_0^inftyint_0^infty e^-p(u+v)f(u)g(v) du dv ,,, , ,,, textu+v=t ,, , ,, textv=x\
        &= int_0^inftyint_x^infty e^-ptf(t-x)g(x) dt dx ,, , ,, textchanging order of integration\
        &= int_0^infty e^-ptBig[int_0^tf(t-x)g(x) dx Big] dt \
        &= cal L(f*g)(t)
        endalign







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 2 hours ago

























        answered 4 hours ago









        Nosrati

        25k62052




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