Is it possible to eliminate a variable from these two differential equations?

Question

I have two second order differential equations in three variables that I need to use to establish a relation between two of the variables.

The equations are

$\alpha$ $ x $ + $\beta$$y$ = $A$$(cx + x'')$$ $ + $\gamma$r

$\alpha$ $ x $ + $\beta$$y$ = $B$$(cx + x'')$$ $ + $\delta$r

where $x$, $y$, $r$ are functions of $z$, and $\alpha$, $\beta$, $\gamma$, $\delta$, $A$, $B$ are constants, and $x''$ = $\frac{\partial^2x}{\partial^2z}$.

I need to get a relation between $y$ and $r$ by eliminating $x$ from the equations, but cannot manage to do this. Is it actually possible to eliminate $x$ from these equations?

Any help would be greatly appreciated.

Answer 1

$$ \cases{B\alpha x + B\beta y = B A(cx + x'') + B\gamma r \\ A\alpha x + A\beta y = AB(cx + x'') + A\delta r }$$

subtracting the second from the first ode we have

$$ \alpha(B-A)x+\beta(B-A)y = (B\gamma-A\delta)r $$

and now if $A\ne B$

$$ x = \left(\frac{B\gamma-A\delta}{B-A}\right)r-\beta y $$

after that we can substitute the found $x$ into the odes.

Answer 2

[If $c$ isn't constant this doesn't work.]

Rearranging your first Eqn gives \begin{align} x''+(c-\frac{\alpha}{A})x=\frac{\beta y-\gamma r}{A}\equiv f(z) \end{align} Letting \begin{align} u&=\kappa x+x'\\ u'&=\kappa x'+x'', \end{align} and substituting into your ODE gives that \begin{align} u'+\frac{Ac-\alpha}{A\kappa}u-\left[\frac{Ac-\alpha}{A\kappa}+\kappa\right]x'=f(z), \end{align} define $\kappa$ such that your $x'$ term goes to zero. . . \begin{align} u'-\kappa u=f(z), \end{align} using integrating factor arrive at the solution for $u$ \begin{align} u(z)=e^{\kappa z}\left(c_1+\int e^{-\kappa z}f(z)\mathrm dz\right)\equiv f_2(z). \end{align} Then solve for $x$ using the definition of $u$ \begin{align} x(z)=e^{-\kappa z}\left(c_2+\int e^{\kappa z}f_2(z)\mathrm dz\right), \end{align} or written out completely, \begin{align} x(z)=e^{-\kappa z}\left(c_2+\int\left[ e^{2\kappa z}\left(c_1+\int e^{-\kappa z}\left[\frac{\beta y-\gamma r}{A}\right]\mathrm dz\right)\right]\mathrm dz\right). \end{align} I hope this helps.

Pro tip: You can use this method to solve linear, non-homogeneous, constant-coefficient ODE's of any order!