Suppose
~a_~n fract{d^~n~y,d~x^~n} + ... + ~a_1 fract{d~y,d~x} + ~a_0 ~y _ = _ ~f ( ~x )
is an ~n^~{th} order linear differential equation (#~{l.d.e.}). The equation is #~{homogenous} _ if _ ~f ( ~x ) = 0 , otherwise it is #~{inhomogenous}.
~a fract{d^2~y,d~x^2} + ~b fract{d~y,d~x} + ~c ~y _ = _ 0 _ _ _ _ _ _ ~a, ~b, ~c _ constants.
Try _ ~y = ~e^{~m~x} _ then _ ~a ~m^2 ~e^{~m~x} + ~b ~m ~e^{~m~x} + ~c ~e^{~m~x} _ = _ 0 _ _ => _ _ ~a ~m^2 + ~b ~m + ~c _ = _ 0
~m _ = _ fract{-~b +- &sqrt.${ ~b&powtwo. - 4~a~c },2~a}
In general there will be two roots for ~m , _ ~p and ~q say. The general solution of the equation is _ ~y _ = _ ~A ~e ^{~p~x} + ~B ~e ^{~q~x}
The equation, _ ~a ~m^2 + ~b ~m + ~c _ = _ 0 , _ is called the ~#{auxiliary equation} or ~#{indicial equation}.
#{Example 1}
fract{d^2~y,d~x^2} - fract{d~y,d~x} - 2 ~y _ = _ 0
~y = ~e^{~m~x} _ is a solution if _ ~m^2 - ~m - 2 _ = _ 0 _ <=> _ ( ~m + 1 ) ( ~m - 2 ) _ = _ 0
i.e. the general soultion is _ ~y _ = _ ~A ~e ^{-~x} + ~B ~e ^{2~x}
#{Example 2}
fract{d^2~y,d~x^2} - ~k^2 ~y _ = _ 0
~y = ~e^{~m~x} _ is a solution if _ ~m^2 - ~k^2 _ = _ 0 _ <=> _ ~m _ = _ +- ~k .
The general soultion is _ ~y _ = _ ~A ~e ^{~k ~x} + ~B ~e ^{- ~k ~x}
#{Example 3}
fract{d^2~y,d~x^2} + ~k^2 ~y _ = _ 0
~y = ~e^{~m~x} _ is a solution if _ ~m^2 + ~k^2 _ = _ 0 _ <=> _ ~m _ = _ +- ~i ~k . _ The general soultion is
~y _ = _ ~A ~e ^{~i ~k ~x} + ~B ~e ^{- ~i ~k ~x} _ = _ ( ~A + ~B ) cos ~k ~x _ + _ ~i ( ~A - ~B ) sin ~k ~x
Note that this is a more general version of the solution , we found earlier, (where ~k = 1). If the initial conditions require that the solution be real, then the general (real) solution is :
~y _ = _ ~C cos ~k ~x _ + _ ~D sin ~k ~x , _ _ ~C, ~D &in. &reals.
where _ ~A = ^~C/_2 - ~i ^~D/_2 _ and _ ~B = ^~C/_2 + ~i ^~D/_2 . _ Of course this is an additional constraint on the constants.
If _ ~b&powtwo. = 4~a~c _ then the two roots will be equal: _ ~p = ~q.
So the 'general' solution becomes _ ~y _ = _ ~A ~e ^{~p~x} + ~B ~e ^{~p~x} _ = _ ~C ~e ^{~p~x} , _ i.e. only one constant.
But in this case _ ~y = ~x ~e ^{~p~x} _ is also a solution, since _ ~p = -~b ./ 2~a , _ and
~y _ = _ ~x ~e ^{~p~x} _ _ => _ _ ~y' _ = _ ~x ~p~e ^{~p~x} + ~e ^{~p~x} = ( ~p ~x + 1 ) ~e ^{~p~x}
=> _ _ _ _ ~y'' _ = _ ~p ~e ^{~p~x} + ( ~p ~x + 1 ) ~p ~e ^{~p~x} _ = _ ( ~p^2 ~x + 2 ~p ) ~e ^{~p~x}
So substituting these values in the differential equation:
~a ~y'' + ~b ~y' + ~c ~y _ = _ ( ( ~a ~p^2 + ~b ~p + ~c ) ~x + 2 ~a ~p + ~b ) ~e ^{~p~x}
but _ ~a ~p^2 + ~b ~p + ~c _ = _ 0 , _ and _ 2 ~a ~p _ = _ 2 ~a # -~b ./ 2 ~a _ = _ -~b , _ proving that _ ~y = ~x ~e ^{~p~x} _ is a solution.
The general solution is therefore: _ _ _ ~y _ = _ ~A ~e ^{~p~x} + ~B ~x ~e ^{~p~x}
#{Example 4}
fract{d^2~y,d~x^2} + 2 fract{d~y,d~x} + ~y _ = _ 0
The auxiliary equation is _ ~m^2 + 2~m + 1 _ = _ 0 , _ i.e. _ ( ~m - 1 )^2 _ = _ 0 _ _ => _ _ ~p = ~q = 1
The general solution is therefore: _ _ _ ~y _ = _ ~A ~e ^~x + ~B ~x ~e ^~x
By dividing through by ~a, a second order equation can be written in the form
fract{d^2~y,d~x^2} + 2~d fract{d~y,d~x} + ~k ~y _ = _ 0
If _ ~k > 0, _ and _ ~d ^2 < ~k _ then the roots of the auxiliary equation are imaginary, and the general solution is
~y _ = _ ~A exp \{ ( -~d + ~i &sqrt.${ ~k - ~d &powtwo. } _ ) ~x \} _ + _ ~B exp \{ ( -~d - ~i &sqrt.${ ~k - ~d &powtwo. } _ ) ~x \}
_ _ = _ ~e^{-~d~x} [ ~A exp \{ ~i ~x &sqrt.${ ~k - ~d &powtwo. } \} _ + _ ~B exp \{ - ~i ~x &sqrt.${ ~k - ~d &powtwo. } \} ]
_ _ = _ ~e^{-~d~x} [ ( ~A + ~B ) cos ( ~x &sqrt.${ ~k - ~d &powtwo. } ) _ + _ ( ~A - ~B ) ~i sin ( ~x &sqrt.${ ~k - ~d &powtwo. } ) ]
In this case the general ~{real valued} solution is
~y _ = _ ~e^{-~d~x} [ ~C cos ( ~x &sqrt.${ ~k - ~d &powtwo. } ) _ + _ ~D sin ( ~x &sqrt.${ ~k - ~d &powtwo. } ) ]
_ _ = _ ~e^{-~d~x} _ ~E _ sin ( ~x &sqrt.${ ~k - ~d &powtwo. } _ + _ &phi. ) , _ _ where _ ~E = &sqrt.${ ~C &powtwo. + ~D &powtwo. } , _ &phi. = sin^{-1} ~C / ~E
If _ ~k > 0, _ and _ ~d ^2 > ~k _ then the roots of the auxiliary equation are real, and the general solution is
~y _ = _ ~A exp \{ ( -~d + &sqrt.${ ~d &powtwo. - ~k } _ ) ~x \} _ + _ ~B exp \{ ( -~d - &sqrt.${ ~d &powtwo. - ~k } _ ) ~x \}
_ _ = _ ~e^{-~d~x} [ ~A exp \{ ~x &sqrt.${ ~d &powtwo. - ~k } \} _ + _ ~B exp \{ - ~x &sqrt.${ ~d &powtwo. - ~k } \} ]
If _ ~k > 0, _ and _ ~d ^2 = ~k _ then the roots are concurrent, and the general solution is
~y _ = _ ~e^{-~d~x} [ ~A + ~B~x ]
We can generalize the above by stating that the equation
~a_~n fract{d^~n~y,d~x^~n} + ... + ~a_1 fract{d~y,d~x} + ~a_0 ~y _ = _ 0
will have a general solution _ _ ~A_1 exp ( ~m_1 ~x ) + ... + ~A_~n exp ( ~m_~n ~x ) , _ _ where _ ~m_1, ... ~m_~n _ are the separate roots of the auxiliary equation
~a_1 ~m_1^2 + ... + ~a_~n ~m_~n^2 _ = _ 0
If this equation does not have ~n separate roots, say for example _ ~m_~i = ~m_~j , _ then the general solution becomes:
~A_1 exp ( ~m_1 ~x ) + ... + ~A_~i exp ( ~m_~i ~x ) + ... + ~A_~j ~x exp ( ~m_~i ~x ) + ... + ~A_~n exp ( ~m_~n ~x )
and if _ ~m_~i = ~m_~j = ~m_~k :
~A_1 exp ( ~m_1 ~x ) + ... + ~A_~i exp ( ~m_~i ~x ) + ... + ~A_~j ~x exp ( ~m_~i ~x ) + ... + ~A_~k ~x^2 exp ( ~m_~i ~x ) + ... + ~A_~n exp ( ~m_~n ~x )
etcetera.