Simplifying two-loop RLC circuit equations for Laplace transform

[formulajoe]

there are two loops in an electrical circuit. I've got two equations and two unknowns.
here are the equations

v(t) = I1*R1 + 1/c*int(I1*dt) - 1/c*int(I2*dt) - first loop
0 = LdI2/dt + R2*I2 - 1/c*int(I1*dt) + 1/c*int(I2*dt) - second loop

the capacitor is in the branch between the two loops.

Im drawing a blank on this. I need to simplify into one equation so I can find the laplace transform of the one equation.
the goal is to find the step response of the system using matlab.

 

[quantumdude]

You'll want to differentiate both equations first, to get rid of those god awful integrals.

I need to simplify into one equation so I can find the laplace transform of the one equation.

There's nothing wrong with taking the Laplace transform first and then simplifying.

 

[formulajoe]

can i get a confirmation that i set up those differential equations properly? I am having some trouble with this still. if i differentiate, than i get derivative of the input. I am trying to find the output voltage which is across the R2 resistor.

 

[formulajoe]

im still having trouble with this problem. I've tried multiple methods including differentiation and solving for a variable and plugging it into the first equation but I am still where i started the other day. I am trying to find an equation that relates the output voltage across R2 to the input. I am confident my beginning equations are correct, but i don't know where to go from there.