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    I dont understand why y=e^mx is the general solution of a 2nd ODE. Also, why do we substitute in the 1st and second derivatives of y=e^mx into the 2ND ODE to find the auxiliary eqn?
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    For a given second order ODE with constant coefficients:

     ay^{''}(x) + by^{'}(x) + cy(x) = 0, where  a, b, c \in \mathbb{C},

     \therefore cy(x) = -ay^{''}(x) - by^{'}(x).

    Notice the above states that the addition of constant multiples of  y^{''}(x) and  y^{'}(x) are equal to  cy(x) .

    ----------------------------------------------------------------------------------------------------------------

    The exponential term  e^{ax} (where a is a constant), is useful here as:

    - Derrivatives of it are just a constant multiple of the exponential term,

    -  e^{ax} = 0, only where x goes to negative infinity.

    So we propose  y(x) = e^{ax} as a solution to the ODE.

    However, what is value a? Substitute  y^{'}(x) and  y^{''}(x) into the ODE to find this.

    ----------------------------------------------------------------------------------------------------------

    (Note the general solution is normally of the form  y(x) = Ae^{a_{1}x} + Be^{a_{2}x}, as a superposition of the solutions to the second order ODE but if you have a single solution for a, then the general solution is slightly different).

    This is the general situation, if anyone wants to correct this then please post.
 
 
 
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