# The Extended Galerkin Method for Approximate Solutions of Nonlinear Vibration Equations
## 3 Application Examples
### The Van Der Pol Equation  
As a popular nonlinear vibration equation different from the Duffing equation, van der Pol equation is also a typical test problem for solution techniques and accuracy [27]. The standard form of van der Pol equation is [9; 28]  
\[\ddot{u}+u=\epsilon\big{(}1-u^{2}\big{)}\dot{u}, \tag{20}\]  
where \(\epsilon\) is a small parameter in this study. The approximate displacement of the first two frequencies is assumed to be  
\[u=A_{1}^{s}\sin\omega t+A_{1}^{c}\cos\omega t+A_{3}^{s}\sin 3\omega t+A_{3}^{c }\cos 3\omega t\, \tag{21}\]  
where \(A\big{(}A_{1}^{s},A_{1}^{c},A_{3}^{s},A_{3}^{c}\big{)}\) and \(\omega\) are amplitudes and frequency to be determined. Now, the general form of the nonlinear vibration equation with the approximate solution in Equation (21) is written as  
\[ \begin{split} N&=A_{1}^{s}\big{(}1-\omega^{2} \big{)}\sin\omega t+A_{1}^{c}\big{(}1-\omega^{2}\big{)}\cos\omega t+A_{3}^{s} \big{(}1-9\omega^{2}\big{)}\sin 3\omega t\\ &+A_{3}^{c}\big{(}1-9\omega^{2}\big{)}\cos 3\omega t-\epsilon \big{[}1-\big{(}A_{1}^{s}\sin\omega t+A_{1}^{c}\cos\omega t+A_{3}^{s}\sin 3 \omega t\\ &+A_{3}^{c}\cos 3\omega t\ \big{)}^{2}\big{]}\omega\big{(}A_{1}^{s}\cos \omega t-A_{1}^{c}\sin\omega t+3A_{3}^{s}\cos 3\omega t-3A_{3}^{c}\sin 3\omega t\ \big{)}.\end{split}  \tag{22}\]  
Applying the general procedure of the extended [PERSON] method, it requires  
\[ \begin{split}\int_{0}^{T}N(A,\ \omega,\epsilon,t)\sin\omega t \text{d}t=0,\qquad\int_{0}^{T}N(A,\ \omega,\epsilon,t)\cos\omega t\text{d}t=0,\\ \int_{0}^{T}N(A,\ \omega,\epsilon,t)\sin 3\omega t\text{d}t=0,\ \ \int_{0}^{T}N(A,\ \omega, \epsilon,t)\cos 3\omega t\text{d}t=0.\end{split}  \tag{23}\]Substituting Equation (22) for Equation (23), it yields  
\[ \begin{split}\big{(}A_{1}^{s}\big{)}^{2}\big{(}A_{1}^{c}-A_{3}^{c} \big{)}\varepsilon\omega+A_{1}^{c}\Big{\{}\big{(}A_{1}^{c}\big{)}^{2}+A_{1}^{c}A _{3}^{c}+2\Big{[}-2+\big{(}A_{3}^{c}\big{)}^{2}+\big{(}A_{3}^{c}\big{)}^{2}\Big{]} \Big{\}}\varepsilon\omega\\ +2A_{1}^{s}\big{(}-2+A_{1}^{c}A_{3}^{s}\varepsilon\omega+2\omega^{2} \big{)}=0,\\ \big{(}A_{1}^{c}\big{)}^{2}\big{(}A_{1}^{s}+A_{3}^{s}\big{)} \varepsilon\omega+A_{1}^{s}\Big{\{}\big{(}A_{1}^{s}\big{)}^{2}-A_{1}^{s}A_{3}^ {s}+2\Big{[}-2+\big{(}A_{3}^{c}\big{)}^{2}+\big{(}A_{3}^{c}\big{)}^{2}\Big{]} \Big{\}}\varepsilon\omega\\ -2A_{1}^{c}\big{(}-2+A_{1}^{c}A_{3}^{c}\varepsilon\omega+2\omega^{2} \big{)}=0,\\ 4A_{3}^{s}-\Big{\{}\big{(}A_{1}^{c}\big{)}^{3}-3A_{1}^{c}\big{(} A_{1}^{s}\big{)}^{2}+6A_{3}^{c}\big{(}A_{1}^{c}\big{)}^{2}+3A_{3}^{c}\big{[}-4+ 2\big{(}A_{1}^{s}\big{)}^{2}+\big{(}A_{3}^{c}\big{)}^{2}+\big{(}A_{3}^{s}\big{)} ^{2}\Big{]}\Big{\}}\varepsilon\omega\\ -36A_{3}^{s}\omega^{2}=0.\end{split}  \tag{24}\]  
Since van der Pol equation is dominantly even by nature, the focus is on the even function solution of time through the elimination of coefficients accordingly. From the first one of Equation (24), it is clear that if higher-order coefficients vanish, simple solutions are \(A_{1}^{c}=2\) and \(A_{1}^{s}=0\), which are known from approximate solutions of other methods [9].