Consider the following BVP : $$y''''=\lambda y+1$$ with the boundary conditions $y(0)=y'(0)=0$ and $y''(1)=y'''(1)=0$. Reduce the BVP to its corresponding Fredholm integral equation using Green's function technique.
What I am going to ask here is a fairly unusual method, since I didn't find it in any textbook till date. We generally use the Green function technique in two types. Firstly if it is of a second order equation, namely $p_0(x)y''+p_1(x)y'+p_2y=0$ with $a_1y(a)+a_2y'(a)=b_1$ and $a_3y(b)+a_4y'(b)=b_2$ then we know that Green's function is given by $$\displaystyle{ G(x,t)= \begin{cases} -\frac{1}{A}u(x)v(t) & \text{if} \ \ a\leq x<t \\ -\frac{1}{A}u(t)v(x) & \text{if} \ \ t< x\leq b \\ \end{cases} }$$ where $A$ is given by the Abel's formula $\displaystyle{u(t)v'(t)-u'(t)v(t)=\frac{A}{p_0(t)}}$. $u(x)$ will satisfy the first boundary condition and $v(x)$ wil satisfy the second boundary condition as well as the differential equation given above. The second type, which is the most rigorous one, uses the properties of Green's function to establish the Green's function. If we return to our first problem, we can take the Green's function for the homogeneous part $y''''=0$ as $$\displaystyle{ G(x,t)= \begin{cases} A_1+A_2x+A_3x^2+A_4x^3 & \text{if} \ \ 0\leq x<t \\ B_1+B_2x+B_3x^2+B_4x^3 & \text{if} \ \ t< x\leq 1 \\ \end{cases} }$$ Now we use for the second method that $G(x,t)$ will satisfy the boundary conditions, its first and second derivative continuity, and third order derivative jump to form a system of eight equations, which on solving will give us the eight constants in $G(x,t)$ and determine the Green's function. But this involves a huge calculation in this regard. So I want to follow the first method using Abel's formula as mentioned above. But in every textbook, this method is applicable only if the equation is of second order precisely. I want to know if there is a generalization of this method (Abel's formula) for a BVP of order greater than or equal to $3$. I calculated that the Wronskian for this system will be $$W(u,v,w,z)=-2t^3+6t^2$$ where I have taken $u(x)=x,v(x)=x^2,w(x)=\frac{x^3}{6}-\frac{x^2}{2}$ and $z(x)=x^3$, all of which satisfies the four individual boundary conditions repectively. But I could not form the Green's function afterwards, as described above in first method. I want a reference if this has been discussed to apply this technique anywhere in history of integral equations. Thanks for your time.
Edit : There is a slight mistake in the choice of $z(x)$ in the last Wronskian I tried to attempt, as it doesn't satisfy $z'''(1)=0$. But at this moment, I'm unable to find one such $z(x)$, from the solution basis. So please ignore that choice above. I am working on it.