General solution of higher order differential equations
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I'm trying to solve this equation:
$$y''' -y'' +y' -y =0.$$
I got the characteristic equation:
$$r^3 -r^2 +r -1 =0$$
and factored by grouping to get:
$$(r+1)(r+1)(r-1) =0$$
so that the roots are $r=1$ and $r=-1$ (repeated).
The problem I have now is getting the general solution - since one of the roots is repeated, would the general solution be
$$y(x) =c_1e^{-x} +c_2e^{-x} +c_3e^x,$$
$$y(x) =c_1e^{-x} +c_2xe^{-x} +c_3e^x,$$
or
$$y(x) =c_1e^{-x} +c_2xe^{-x} +c_3x^2e^x,$$
or something different? I'm completely lost so any help would be very much appreciated. Thanks!
ordinary-differential-equations
edited Nov 29 '18 at 7:36
Rócherz
2,7762721
asked Nov 29 '18 at 7:27
seralinseralin
61
$endgroup$
add a comment |
$begingroup$
I'm trying to solve this equation:
$$y''' -y'' +y' -y =0.$$
I got the characteristic equation:
$$r^3 -r^2 +r -1 =0$$
and factored by grouping to get:
$$(r+1)(r+1)(r-1) =0$$
so that the roots are $r=1$ and $r=-1$ (repeated).
The problem I have now is getting the general solution - since one of the roots is repeated, would the general solution be
$$y(x) =c_1e^{-x} +c_2e^{-x} +c_3e^x,$$
$$y(x) =c_1e^{-x} +c_2xe^{-x} +c_3e^x,$$
or
$$y(x) =c_1e^{-x} +c_2xe^{-x} +c_3x^2e^x,$$
or something different? I'm completely lost so any help would be very much appreciated. Thanks!
ordinary-differential-equations
edited Nov 29 '18 at 7:36
Rócherz
2,7762721
asked Nov 29 '18 at 7:27
seralinseralin
61
$endgroup$
$begingroup$
$(a+bx)e^{-x}+ce^x$
$endgroup$
– Lord Shark the Unknown
Nov 29 '18 at 7:28
4
$begingroup$
Are you sure about the factorization? I see $(r-1)(r^2+1)=0$, without repeated roots.
$endgroup$
– LutzL
Nov 29 '18 at 7:30
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Nov 29 '18 at 7:34
add a comment |
$begingroup$
I'm trying to solve this equation:
$$y''' -y'' +y' -y =0.$$
I got the characteristic equation:
$$r^3 -r^2 +r -1 =0$$
and factored by grouping to get:
$$(r+1)(r+1)(r-1) =0$$
so that the roots are $r=1$ and $r=-1$ (repeated).
The problem I have now is getting the general solution - since one of the roots is repeated, would the general solution be
$$y(x) =c_1e^{-x} +c_2e^{-x} +c_3e^x,$$
$$y(x) =c_1e^{-x} +c_2xe^{-x} +c_3e^x,$$
or
$$y(x) =c_1e^{-x} +c_2xe^{-x} +c_3x^2e^x,$$
or something different? I'm completely lost so any help would be very much appreciated. Thanks!
ordinary-differential-equations
edited Nov 29 '18 at 7:36
Rócherz
2,7762721
asked Nov 29 '18 at 7:27
seralinseralin
61
$endgroup$
I'm trying to solve this equation:
$$y''' -y'' +y' -y =0.$$
I got the characteristic equation:
$$r^3 -r^2 +r -1 =0$$
and factored by grouping to get:
$$(r+1)(r+1)(r-1) =0$$
so that the roots are $r=1$ and $r=-1$ (repeated).
The problem I have now is getting the general solution - since one of the roots is repeated, would the general solution be
$$y(x) =c_1e^{-x} +c_2e^{-x} +c_3e^x,$$
$$y(x) =c_1e^{-x} +c_2xe^{-x} +c_3e^x,$$
or
$$y(x) =c_1e^{-x} +c_2xe^{-x} +c_3x^2e^x,$$
or something different? I'm completely lost so any help would be very much appreciated. Thanks!
ordinary-differential-equations
ordinary-differential-equations
edited Nov 29 '18 at 7:36
Rócherz
2,7762721
asked Nov 29 '18 at 7:27
seralinseralin
61
edited Nov 29 '18 at 7:36
Rócherz
2,7762721
asked Nov 29 '18 at 7:27
seralinseralin
61
edited Nov 29 '18 at 7:36
Rócherz
2,7762721
edited Nov 29 '18 at 7:36
Rócherz
2,7762721
edited Nov 29 '18 at 7:36
Rócherz
2,7762721
2,7762721
asked Nov 29 '18 at 7:27
seralinseralin
61
asked Nov 29 '18 at 7:27
seralinseralin
61
asked Nov 29 '18 at 7:27
seralinseralin
61
61
$begingroup$
$(a+bx)e^{-x}+ce^x$
$endgroup$
– Lord Shark the Unknown
Nov 29 '18 at 7:28
4
$begingroup$
Are you sure about the factorization? I see $(r-1)(r^2+1)=0$, without repeated roots.
$endgroup$
– LutzL
Nov 29 '18 at 7:30
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Nov 29 '18 at 7:34
add a comment |
$begingroup$
$(a+bx)e^{-x}+ce^x$
$endgroup$
– Lord Shark the Unknown
Nov 29 '18 at 7:28
4
$begingroup$
Are you sure about the factorization? I see $(r-1)(r^2+1)=0$, without repeated roots.
$endgroup$
– LutzL
Nov 29 '18 at 7:30
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Nov 29 '18 at 7:34
$begingroup$
$(a+bx)e^{-x}+ce^x$
$endgroup$
– Lord Shark the Unknown
Nov 29 '18 at 7:28
$begingroup$
$(a+bx)e^{-x}+ce^x$
$endgroup$
– Lord Shark the Unknown
Nov 29 '18 at 7:28
4
4
$begingroup$
Are you sure about the factorization? I see $(r-1)(r^2+1)=0$, without repeated roots.
$endgroup$
– LutzL
Nov 29 '18 at 7:30
$begingroup$
Are you sure about the factorization? I see $(r-1)(r^2+1)=0$, without repeated roots.
$endgroup$
– LutzL
Nov 29 '18 at 7:30
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Nov 29 '18 at 7:34
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Nov 29 '18 at 7:34
add a comment |
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$begingroup$
$(a+bx)e^{-x}+ce^x$
$endgroup$
– Lord Shark the Unknown
Nov 29 '18 at 7:28
4
$begingroup$
Are you sure about the factorization? I see $(r-1)(r^2+1)=0$, without repeated roots.
$endgroup$
– LutzL
Nov 29 '18 at 7:30
$begingroup$
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– José Carlos Santos
Nov 29 '18 at 7:34