I have read this:Systems of Differential Equations and higher order Differential Equations and I am trying to apply it to the following set of equations but with not much success
$$L_{11} \frac{dI_1}{dt}+L_{12}\frac{dI_2}{dt}+I_1R_1=\varepsilon(t)$$ $$L_{22}\frac{dI_2}{dt}+L_{12}\frac{dI_1}{dt}+I_2R_2=0$$ I isolated $I_2$ from the second equation and took the derivative with respect to $t$ and I got:
$$\frac{dI_2}{dt}=-\frac{1}{R_2}\left(L_{22}\frac{d^2I_2}{dt^2}+L_{12}\frac{d^2I_1}{dt^2}\right)$$
but when I substitute $\frac{dI_2}{dt}$ in the first equation I get:
$$L_{11} \frac{dI_1}{dt}-\frac{L_{12} L_{22}}{R_2}\frac{d^2I_2}{dt^2}-\frac{L_{12}^2}{R_2}\frac{d^2I_1}{dt^2}+I_1R_1=\varepsilon(t)$$
so i take the derivative of the expression for $\frac{dI_2}{dt}$ above but i keep getting higher and higher orders differential equations
