## Appendix A
 \tag{125}\]  
\[\begin{array}{c}(b_{11}-d_{11})\big{(}w^{IV}-l^{2}w^{VI}\big{)}+pb_{11}\Big{[}-3{ w^{IV}}{w^{\prime}}^{2}+\frac{9}{4}w^{IV}{w^{\prime}}^{4}-\frac{9}{4}{w^{\prime}}^{2}{w^{ \prime}}^{3}+9{w^{\prime\prime}}^{w^{\prime\prime}}-l^{2}\Big{(}-3{w^{VI}}{w^{ \prime}}^{2}-12w^{V}{w^{\prime}}^{w^{\prime\prime}}^{-}\\ 6{w^{\prime}}^{2}w^{IV}-6{w^{\prime}}{w^{\prime\prime}}^{w^{\prime}}w^{IV}+\frac{ 9}{4}w^{IV}{w^{\prime}}^{IV}+18{w^{\prime}}^{w^{\prime}}{w^{\prime\prime}}^{3} {w^{\prime}}+27{w^{IV}}^{2}{w^{\prime\prime}}^{2}{w^{\prime}}^{2}+9{w^{ \prime}}^{3}{w^{\prime\prime}}^{w^{\prime}}+54{w^{\prime\prime}}^{3}{w^{ \prime\[\overline{n}_{2}=3D_{11}\int_{0}^{1}\bigl{[}\psi^{\prime\prime}(x)\varphi(x)-l^{2} \psi^{V}(x)\varphi(x)\bigr{]}dx-A_{55}\int_{0}^{1}\big{[}\psi^{\prime}(x)-l^{2} \psi^{\prime\prime\prime}(x)\big{]}\varphi(x)dx\] \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\] \[3\int_{0}^{1}\bigl{[}\varphi^{\prime}(x)\varphi^{\prime\prime}(x) \varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}3\varphi^{\prime\prime}(x) \varphi^{\prime\prime\prime}(x)\varphi^{IV}(x)+\varphi^{\prime}(x)\varphi^{ \prime\prime}(x)\varphi^{\prime\prime}(x)+\] \[2\varphi^{\prime\prime}{}^{2}(x)\varphi^{IV}(x)\bigr{)}\varphi( x)]dx+3\int_{0}^{1}\bigl{[}\varphi^{\prime\prime 2}(x)\varphi^{\prime IV}(x)\varphi(x)-l^{2} \Bigl{(}\varphi^{\prime\prime 2}(x)\varphi^{IV}(x)+2\varphi^{\prime}(x) \varphi^{\prime\prime}(x)\varphi^{IV}(x)+\] \[4\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{V}(x) \bigr{)}+\varphi^{\prime 2}(x)\varphi^{VI}(x)\bigr{]}dx+3\int_{0}^{1}\bigl{[} \varphi^{\prime\prime 3}(x)\varphi(x)-l^{2}\Bigl{(}6\varphi^{\prime\prime}(x) \varphi^{\prime\prime}{}^{2}(x)+\] \[3\varphi^{\prime\prime}{}^{2}(x)\varphi^{\prime IV}(x)\bigr{)} \varphi(x)]dx\bigr{)}+D_{11}\Bigl{\{}3\int_{0}^{1}\Bigl{[}\varphi^{\prime IV}(x) \varphi^{\prime 2}(x)\varphi(x)-l^{2}\bigl{(}\varphi^{\prime}(x)\varphi^{ \prime\prime}(x)+4\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{IV}(x) \bigr{)}+\] \[2\varphi^{\prime\prime}{}^{2}(x)\varphi^{\prime IV}(x)+2\varphi^{ \prime\prime}(x)\varphi^{\prime\prime}(x)\varphi(x)\Bigr{]}dx-9\int_{0}^{1} \Bigl{[}\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime\prime \prime}(x)\varphi(x)-l^{2}\bigl{(}3\varphi^{\prime}(x)\varphi^{\prime\prime 2}(x)+\] \[3\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{IV}(x)+ \varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime}(x)\varphi^{IV}(x )+2\varphi^{\prime\prime}{}^{2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)]dx\bigr{\}}+\] \[E_{11}\Bigl{\{}-\frac{3}{2}\int_{0}^{1}\Bigl{[}\varphi^{\prime 2}(x) \varphi^{IV}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime\prime 2}(x)\varphi^{\prime V}(x)+2 \varphi^{\prime}(x)\varphi^{\prime\prime}(x)+4\varphi^{\prime}(x)\varphi^{ \prime\prime}(x)\varphi^{V}(x)+\] \[\varphi^{\prime 2}(x)\varphi^{VI}(x)\bigr{)}\varphi(x)]dx\bigr{\}}+(K_{3}+2K) \int_{0}^{1}\varphi^{4}(x)dx\] \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]0 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]1 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]2 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]3 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]4 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]5 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]6 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]7 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]8 \[\overline{b}_{3}=B_{11}\Bigl{\{}-3\int_{0}^{1}\Bigl{[}\varphi^{ \prime 2}(x)\varphi^{\prime\prime}(x)\varphi(x)-l^{2}\bigl{(}2\varphi^{\prime \prime}{}^{3}(x)+6\varphi^{\prime}(x)\varphi^{\prime\prime}(x)\varphi^{\prime \prime}(x)+\varphi^{\prime 2}(x)\varphi^{IV}(x)\bigr{)}\varphi(x)\Bigr{]}dx-\]9 \[\overline{b}_{7}=-A_{11}\lambda^{4}\Bigl{(}\int_{0}^{1}\varphi^{ \prime\prime}{}^{4}(x)\varphi(x)dx\Bigr{)}\Bigl{\{}\int_{0}^{1}\Bigl{[} \varphi^{\prime}(x)\varphi^{\prime\prime}{}^{2}(x)\varphi(x)-l^{2}\bigl{(} \varphi^{\prime\prime\prime}(x)\varphi^{\prime\prime}(x)+4\varphi^{\prime