Iconic_panda
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when solving 2nd order DE, can we jump straight into the auxiliary equation or do we have to induce the auxiliary equation by firstly assuming the answer must be in a form of Ae^kx?
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RDKGames
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(Original post by Iconic_panda)
when solving 2nd order DE, can we jump straight into the auxiliary equation or do we have to induce the auxiliary equation by firstly assuming the answer must be in a form of Ae^kx?
Well if you assume that the C.F. of y''+py'+qy=r(x) has the form of y=Ae^{mx} then the following steps of:

y' = Ame^{mx}

y''=Am^2 e^{mx}

\Rightarrow Am^2 e^{mx} + Apme^{mx} + Aqe^{mx} = 0

\Rightarrow Ae^{mx}(m^2+pm+q) = 0

\Rightarrow m^2+pm+q=0

can all be skipped to just say the last line since at A-Level this is the only function you really work with on 2nd order ODE's and this will happen every time you use it.
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Iconic_panda
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thank you
(Original post by RDKGames)
Well if you assume that the C.F. of y''+py'+qy=r(x) has the form of y=Ae^{mx} then the following steps of:

y' = Ame^{mx}

y''=Am^2 e^{mx}

\Rightarrow Am^2 e^{mx} + Apme^{mx} + Aqe^{mx} = 0

\Rightarrow Ae^{mx}(m^2+pm+q) = 0

\Rightarrow m^2+pm+q=0

can all be skipped to just say the last line since at A-Level this is the only function you really work with on 2nd order ODE's and this will happen every time you use it.
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