Solving a system of second order ODEs with initial conditions.

Question

Essentially, this is the tail end of a problem I have. The result of which gave two second order ODEs:

$u'' = λv$

$v'' = λu$

Where

$u(0) = 0$, $u(1) = 1$, $v(0) = 0$, $v(0) = 0$

and

$\int_{0}^{1}uv \,dx=1$

(The integral portion I am unsure if it is pertinent to the solution of the ODEs, but the boundary conditions I am fairly certain are. λ is a constant, and is the result of finding the Euler-Lagrange equations for $\min\limits_{u,v}\int_{0}^{1}[(u')^2+(v')^2] \,dx=1$, this is more or less just extra information for context)

Basically I don't know how to solve the ODEs, if they're even solvable at all. Any suggestions or help would be really appreciated.

Answer 1

$$u'' = \lambda v$$ $$v'' = \lambda u$$ Add both differential equations and solve: $$(u+v)''=\lambda (u+v)$$

Answer 2

Hint: Introduce new variables $$u_{\pm}~:=~u\pm v.$$ Then OP's 2 coupled ODEs (and functional) separate.