Methods for converting PWM to +/- 10v Signal
Something like this should work well. Parts are just for example, optimization will take more work (and information). 







simulate this circuit &ndash; Schematic created using CircuitLab



Edit: 

I've been asked for analysis of the amplifier (gain = +2 for switch closed, -2 for switch open). 

First observe that the op-amp inputs are always at Vin/2 at balance due to R3/R5. 

(Note also that they are always positive since Vin varies from 0V to 5V, so M1 always sees positive voltage on the drain- 0 to 2.5V for 0 to 5V in)  

Switch M1 open (off), we have current (Vin - Vin/2)/R2 flowing through R2, so - (Vout - Vin/2)/R1 must be flowing through R1, using KCL.  

Solve for Vout/Vin = 1/2 - R1/(2*R2) = -2.00 for R1 = 5*R2



With the switch closed (assume M1 = 0\$\Omega\$) we have an added term and using KCL again, the result is:

Vin/(2*R2) - Vin/(2*R4) + (Vout -Vin/2)/R1 =0 

Solve for Vout/Vin = 1/2 +R1/(2*R4) - R1/(2*R2)  = +2.00 for R1 = 5*R2 R4 = (25/40) * R2.