# Revision history [back]

By solving against Vo only, you do as if every other symbols involved in the equations were constants. However, Vi, Vm, Vx are also unknown.

Imagine that the solver would return [Vo == -A*Vm] as a solution, i bet you will not be very happy.

So, what you have to do is to put all unknowns, and let only the constants outside:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vi, Vm, Vx])
[[Vo == (A*R3*r1 + (A*C*R3*r1*s + A*r1)*R2)/(A*R2 - R3), Vi == -((C*R3*r1*s + A*r1 + r1)*R1 + (C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vm == -((C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vx == r1]]


Now, as you can see, there is one degree of freedom : Vx can take any value, so Sage added a free parameter r1 and gave you all the solutions using this free parameter r1.

Now, i see that you want to solve for Vo/Vi, so let me assume that you want Vi being the free parameter, not Vx.

Hence, the way to do is to solve as if Vi was also a constant:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vm, Vx])
[[Vo == -((A*C*R3*s + A)*R2 + A*R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vm == ((C*R3*s + 1)*R2 + R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vx == -(A*R2 - R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3)]]


By solving against Vo only, you do as if every other symbols involved in the equations were constants. However, Vi, Vm, Vx are also unknown.

Imagine that the solver would return [Vo == -A*Vm] as a solution, i bet you will not be very happy.

So, what you have to do is to put all unknowns, and let only the constants outside:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vi, Vm, Vx])
[[Vo == (A*R3*r1 + (A*C*R3*r1*s + A*r1)*R2)/(A*R2 - R3), Vi == -((C*R3*r1*s + A*r1 + r1)*R1 + (C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vm == -((C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vx == r1]]


Now, as you can see, there is one degree of freedom : Vx can take any value, so Sage added a free parameter r1 and gave you all the solutions using this free parameter r1.

Now, i see in your workaround that you want to solve for Vo/Vi, so let me assume that you want Vi being the free parameter, not Vx.

Hence, the way to do is to solve as if Vi was also a constant:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vm, Vx])
[[Vo == -((A*C*R3*s + A)*R2 + A*R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vm == ((C*R3*s + 1)*R2 + R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vx == -(A*R2 - R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3)]]


By solving against Vo only, you do as if every other symbols involved in the equations were constants. However, Vi, Vm, Vx are also unknown.

Imagine that the solver would return [Vo == -A*Vm] as a solution, i bet you will not be very happy.

So, what you have to do is to put all unknowns, and let only the constants outside:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vi, Vm, Vx])
[[Vo == (A*R3*r1 + (A*C*R3*r1*s + A*r1)*R2)/(A*R2 - R3), Vi == -((C*R3*r1*s + A*r1 + r1)*R1 + (C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vm == -((C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vx == r1]]


Now, as you can see, there is one degree of freedom : Vx can take any value, so Sage added a free parameter r1 and gave you all the solutions using the constants and this free parameter r1.

Now, i see in your workaround that you want to solve for Vo/Vi, so let me assume that you want Vi being the free parameter, not Vx.

Hence, the way to do is to solve as if Vi was also a constant:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vm, Vx])
[[Vo == -((A*C*R3*s + A)*R2 + A*R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vm == ((C*R3*s + 1)*R2 + R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vx == -(A*R2 - R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3)]]


By solving against Vo only, you do as if every other symbols involved in the equations were constants. However, Vi, Vm, Vx are also unknown.

Imagine that the solver would return [Vo == -A*Vm] as a solution, i bet you will not be very happy.

So, what you have to do is to put all unknowns, and let only the constants outside:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vi, Vm, Vx])
[[Vo == (A*R3*r1 + (A*C*R3*r1*s + A*r1)*R2)/(A*R2 - R3), Vi == -((C*R3*r1*s + A*r1 + r1)*R1 + (C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vm == -((C*R3*r1*s + r1)*R2 + R3*r1)/(A*R2 - R3), Vx == r1]]


Now, as you can see, there is one degree of freedom : Vx can take any value, so Sage added a free parameter r1 and gave you all the solutions using the constants and this free parameter r1.

Now, i see in your workaround that you want to solve for Vo/Vi, so let me assume that you want Vi being the free parameter, not Vx.

Hence, the way to do is to you can solve as if Vi was is also a constant:

sage: solve([eq_at_Vm, eq_at_Vx, eq_at_Vo],[Vo, Vm, Vx])
[[Vo == -((A*C*R3*s + A)*R2 + A*R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vm == ((C*R3*s + 1)*R2 + R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3), Vx == -(A*R2 - R3)*Vi/((C*R3*s + A + 1)*R1 + (C*R3*s + 1)*R2 + R3)]]