### Author Topic: Inquiry about finding third solution using reduction of order  (Read 2232 times)

#### Hyunmin Jung

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##### Inquiry about finding third solution using reduction of order
« on: October 29, 2014, 08:26:50 AM »
If you are given two solutions that satisfies the ODE.

Without wronskian given, using two solutions and by using reduction of order,

is there a way where you are able to find third solutions to the equation?

I could not really get a clear answer from office hour and the problem I am referring is textbook p.228 #28
« Last Edit: October 29, 2014, 11:40:50 AM by Hyunmin Jung »

#### Victor Ivrii

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##### Re: Inquiry about finding third solution using reduction of order
« Reply #1 on: October 29, 2014, 11:09:18 AM »
If you are given two solutions that satisfies the ODE.

Without wronskian given, using two solutions and by using reduction of order,

is there a way where you are able to find third solutions to the equation?

I could not really get a clear answer from office hour and the problem I am referring is textbook p.228 #18

There is no problem 18 on p 228 (at least in 10th or 9th editions). What order ODE are you talking about?

#### Hyunmin Jung

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##### Re: Inquiry about finding third solution using reduction of order
« Reply #2 on: October 29, 2014, 11:41:04 AM »
sorry, corrected.

#### Victor Ivrii

If we plug $y=zy_1$ we get for $z$ another third order linear homogeneous linear ODE which, however does not contain $z$ without derivatives, so denoting $z'=u$ we get a second order linear homogeneous linear ODE for $u$. But we know one of its linearly independent solutions, namely $u_1=(y_2/y_1)'$ and then plugging $u=vu_1$ we get another second order linear homogeneous linear ODE for $v$ which does not contain $v$ without derivatives so denoting $v'=w$ we get a first order linear homogeneous linear ODE for $u$.
However there is a little shortcut: since $u_1=1$ due to $y_1=t^2$ and $y_2=t^3$ the third order equation for $z$ contains neither $z$ nor $z'$ but only $z''$ and $z'''$ so denoting $z''=w$ we drop to the first order ODE just in one step instead of two.