1. ## \varphi'''-\lambda\varphi=0

$\varphi'''-\lambda\varphi=0$

$\varphi_1(0)=1, \ \varphi_2(0)=0, \ \varphi_3(0)=0$
$\varphi_1'(0)=0, \ \varphi_2'(0)=1, \ \varphi_3'(0)=0$
$\varphi_1''(0)=0, \ \varphi_2''(0)=0, \ \varphi_3''(0)=1$

$m^3-\lambda=0\Rightarrow m^3=\lambda$

$\displaystyle w_k=\lambda^{1/3}\left[\cos\left(\frac{2\pi k}{3}\right)+i\sin\left(\frac{2\pi k}{3}\right)\right]$

Let $\lambda^{1/3}=s$

$\displaystyle\varphi=Ae^{xs}+\exp\left(\frac{-xs}{2}\right)\left[B\cos\left(\frac{xs\sqrt{3}}{2}\right)+C\sin\left( \frac{xs\sqrt{3}}{2}\right)\right]$

$\displaystyle\varphi'=Ase^{xs}-\frac{\exp\left(\frac{-xs}{2}\right)}{2}\left[B\cos\left(\frac{xs\sqrt{3}}{2}\right)+C\sin\left( \frac{xs\sqrt{3}}{2}\right)\right]+\exp\left(\frac{-xs}{2}\right)\left[\frac{Cs\sqrt{3}}{2}\cos\left(\frac{xs\sqrt{3}}{2} \right)-\frac{Bs\sqrt{3}}{2}\sin\left(\frac{xs\sqrt{3}}{2} \right)\right]$

$\displaystyle\varphi''=As^2e^{xs}+\frac{\exp\left( \frac{-xs}{2}\right)}{4}\left[B\cos\left(\frac{xs\sqrt{3}}{2}\right)+C\sin\left( \frac{xs\sqrt{3}}{2}\right)\right]-s\exp\left(\frac{-xs}{2}\right)\left[\frac{Cs\sqrt{3}}{2}\cos\left(\frac{xs\sqrt{3}}{2} \right)-\frac{Bs\sqrt{3}}{2}\sin\left(\frac{xs\sqrt{3}}{2} \right)\right]$ $\displaystyle +\exp\left(\frac{-xs}{2}\right)\left[\frac{-3Bs^2}{4}\cos\left(\frac{xs\sqrt{3}}{2}\right)-\frac{3Cs^2}{4}\sin\left(\frac{xs\sqrt{3}}{2}\righ t)\right]$

$\varphi_1(0)=A+B=1, \ \varphi_2(0)=A+B=0, \ \varphi_3(0)=A+B=0$
$\displaystyle\varphi_1'(0)=sA-\frac{sB}{2}+\frac{Cs\sqrt{3}}{2}=0, \ \varphi_2'(0)=sA-\frac{sB}{2}+\frac{Cs\sqrt{3}}{2}=1, \ \varphi_3'(0)=sA-\frac{sB}{2}+\frac{Cs\sqrt{3}}{2}=0$
$\displaystyle\varphi_1''(0)=s^2A+s^2B-\frac{s^2\sqrt{3}C}{2}=0, \ \varphi_2''(0)=s^2A+s^2B-\frac{s^2\sqrt{3}C}{2}=0, \ \varphi_3''(0)=s^2A+s^2B-\frac{s^2\sqrt{3}C}{2}=1$

$\displaystyle\varphi_1\text{coefficient matrix}=\begin{bmatrix}1&1&0&1\\s&\frac{s}{2}&\fra c{s\sqrt{3}}{2}&0\\s^2&s^2&-\frac{s^2\sqrt{3}}{2}&0\end{bmatrix}\Rightarrow\te xt{rref}=\begin{bmatrix}1&0&0&-\frac{1}{3}\\0&1&0&\frac{4}{3}\\0&0&1&\frac{2\sqrt {3}}{3}\end{bmatrix}$

$\displaystyle\varphi_2\text{coefficient matrix}=\begin{bmatrix}1&1&0&0\\s&\frac{s}{2}&\fra c{s\sqrt{3}}{2}&1\\s^2&s^2&-\frac{s^2\sqrt{3}}{2}&0\end{bmatrix}\Rightarrow\te xt{rref}=\begin{bmatrix}1&0&0&\frac{2}{3s}\\0&1&0&-\frac{2}{3s}\\0&0&1&0\end{bmatrix}$

$\displaystyle\varphi_3\text{coefficient matrix}=\begin{bmatrix}1&1&0&0\\s&\frac{s}{2}&\fra c{s\sqrt{3}}{2}&0\\s^2&s^2&-\frac{s^2\sqrt{3}}{2}&1\end{bmatrix}\Rightarrow\te xt{rref}=\begin{bmatrix}1&0&0&\frac{2}{3s^2}\\0&1& 0&-\frac{2}{3s^2}\\0&0&1&-\frac{2\sqrt{3}}{3s^2}\end{bmatrix}$

However, there is an error some where because the book provides the coefficients for $\varphi_3$ which are:

$\displaystyle A=\frac{1}{3s}, \ B=-\frac{1}{3s}, \ C=-\frac{\sqrt{3}}{3s}$

Therefore, I am not sure about my other coefficients for $\varphi_1, \ \varphi_2$

2. The DE is linear with constant coefficients, so that $\varphi(*)$ is analythic and we can write...

$\displaystyle \varphi(t)= \sum_{n=0}^{\infty} \frac{\varphi^{(n)} (0)}{n!}\ t^{n}$ (1)

But is...

$\varphi^{(n)} (0) = \lambda\ \varphi^{(n-3)} (0)$ (2)

... so that if we set...

$\varphi(0)=a_{0}$

$\varphi^{'}(0)=a_{1}$

$\varphi^{''}(0)=a_{2}$ (3)

... the solution is...

$\displaystyle \varphi(t)= a_{0}\ \sum_{k=0}^{\infty} \frac{\lambda^{k}}{(3k)!}\ t^{3k} + a_{1}\ \sum_{k=0}^{\infty} \frac{\lambda^{k}}{(3k+1)!}\ t^{3k+1} + a_{2}\ \sum_{k=0}^{\infty} \frac{\lambda^{k}}{(3k+2)!}\ t^{3k+2}$ (4)

Kind regards

$\chi$ $\sigma$

3. Originally Posted by chisigma
The DE is linear with constant coefficients, so that $\varphi(*)$ is analythic and we can write...

$\displaystyle \varphi(t)= \sum_{n=0}^{\infty} \frac{\varphi^{(n)} (0)}{n!}\ t^{n}$ (1)

But is...

$\varphi^{(n)} (0) = \lambda\ \varphi^{(n-3)} (0)$ (2)

... so that if we set...

$\varphi(0)=a_{0}$

$\varphi^{'}(0)=a_{1}$

$\varphi^{''}(0)=a_{2}$ (3)

... the solution is...

$\displaystyle \varphi(t)= a_{0}\ \sum_{k=0}^{\infty} \frac{\lambda^{k}}{(3k)!}\ t^{3k} + a_{1}\ \sum_{k=0}^{\infty} \frac{\lambda^{k}}{(3k+1)!}\ t^{3k+1} + a_{2}\ \sum_{k=0}^{\infty} \frac{\lambda^{k}}{(3k+2)!}\ t^{3k+2}$ (4)

Kind regards

$\chi$ $\sigma$
This is probably a great response, but, unfortunately, I don't know how to apply this to the DE.

4. Problem found. $\displaystyle\frac{1}{4}+\frac{-3}{4}=\frac{-1}{2}\neq 1$

$\displaystyle\varphi_1\text{coefficient matrix}=\begin{bmatrix}1&1&0&1\\s&-\frac{s}{2}&\frac{s\sqrt{3}}{2}&0\\s^2&-\frac{s^2}{2}&-\frac{s^2\sqrt{3}}{2}&0\end{bmatrix}\Rightarrow\te xt{rref}=\begin{bmatrix}1&0&0&\frac{1}{3}\\0&1&0&\ frac{2}{3}\\0&0&1&0\end{bmatrix}$

$\displaystyle\varphi_1=\frac{e^{xs}}{3}+\frac{2}{3 }\exp\left(\frac{-xs}{2}\right)\cos\left(\frac{xs\sqrt{3}}{2}\right)$

$\displaystyle\varphi_2\text{coefficient matrix}=\begin{bmatrix}1&1&0&0\\s&-\frac{s}{2}&\frac{s\sqrt{3}}{2}&1\\s^2&-\frac{s^2}{2}&-\frac{s^2\sqrt{3}}{2}&0\end{bmatrix}\Rightarrow\te xt{rref}=\begin{bmatrix}1&0&0&\frac{1}{3s}\\0&1&0&-\frac{1}{3s}\\0&0&1&\frac{1}{s\sqrt{3}}\end{bmatri x}$

$\displaystyle\varphi_2=\frac{e^{xs}}{3s}+\frac{\ex p\left(\frac{-xs}{2}\right)}{3s}\left[-\cos\left(\frac{xs\sqrt{3}}{2}\right)+\sqrt{3}\sin \left(\frac{xs\sqrt{3}}{2}\right)\right]$

$\displaystyle\varphi_3\text{coefficient matrix}=\begin{bmatrix}1&1&0&0\\s&-\frac{s}{2}&\frac{s\sqrt{3}}{2}&0\\s^2&-\frac{s^2}{2}&-\frac{s^2\sqrt{3}}{2}&1\end{bmatrix}\Rightarrow\te xt{rref}=\begin{bmatrix}1&0&0&\frac{1}{3s^2}\\0&1& 0&-\frac{1}{3s^2}\\0&0&1&-\frac{\sqrt{3}}{3s^2}\end{bmatrix}$

$\displaystyle\varphi_3=\frac{e^{xs}}{3s^2}-\frac{\exp\left(\frac{-xs}{2}\right)}{3s^2}\left[\cos\left(\frac{xs\sqrt{3}}{2}\right)+\sqrt{3}\sin \left(\frac{xs\sqrt{3}}{2}\right)\right]$

Find $\varphi(0)=5, \ \varphi'(0)=-3, \ \varphi''(0)=9$

$\displaystyle \varphi(x)=A\left[\frac{e^{xs}}{3}+\frac{2}{3}\exp\left(\frac{-xs}{2}\right)\cos\left(\frac{xs\sqrt{3}}{2}\right) \right]$ $\displaystyle +B\left[\frac{e^{xs}}{3s}+\frac{\exp\left(\frac{-xs}{2}\right)}{3s}\left[-\cos\left(\frac{xs\sqrt{3}}{2}\right)+\sqrt{3}\sin \left(\frac{xs\sqrt{3}}{2}\right)\right]\right]$ $\displaystyle +C\left[\varphi_3=\frac{e^{xs}}{3s^2}-\frac{\exp\left(\frac{-xs}{2}\right)}{3s^2}\left[\cos\left(\frac{xs\sqrt{3}}{2}\right)+\sqrt{3}\sin \left(\frac{xs\sqrt{3}}{2}\right)\right]\right]$

$\varphi(0):A=5$

$\displaystyle\varphi'(0):B=\frac{-54(s-2)}{17s}$

$\displaystyle\varphi''(0):C=\frac{-9(s-36)}{34}$

Are my coefficients correct?