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## Homework Statement

Find the differential equation for VC

Circuit #1

Circuit #2

## Homework Equations

KCL

## The Attempt at a Solution

Circuit #1:

Node VC:

##\frac{V_C-V_{in}}{R_1}+\frac{1}{L_1}\int_{0}^{t}(V_C-V_2)+C_1\dot{V_C}=0##

Node V2:

##\frac{1}{L_1}\int_{0}^{t}(V_2-V_C)+\frac{V_2}{R_2}=0##

Derive NodeVC and solve for V2

##\frac{\dot{V_C}}{R_1}-\frac{\dot{V_{in}}}{R_1}+\frac{V_C}{L_1}-\frac{V_2}{L_1}+C_1\ddot{V_C}=0##

##\frac{\dot{V_C}}{R_1}-\frac{\dot{V_{in}}}{R_1}+\frac{V_C}{L_1}+C_1\ddot{V_C}=\frac{V_2}{L_1}##

##\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}=V_2##

Plug that into node V2 equation

##\frac{1}{L_1}\int_{0}^{t}((\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C})-V_C)+\frac{\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}}=0##

Simplify

##\frac{1}{L_1}\int_{0}^{t}(\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+C_1L_1\ddot{V_C})+\frac{\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}}{R_2}=0##

get rid of integral

##\frac{1}{L_1}(\frac{L_1{V_C}}{R_1}-\frac{L_1{V_{in}}}{R_1}+C_1L_1\dot{V_C})+\frac{\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}}{R_2}=0##

Simplify

##(\frac{{V_C}}{R_1}-\frac{{V_{in}}}{R_1}+C_1\dot{V_C})+\frac{\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}}{R_2}=0##

Multiply everything by R2

##R_2(\frac{{V_C}}{R_1}-\frac{{V_{in}}}{R_1}+C_1\dot{V_C})+\frac{L_1\dot{V_C}}{R_1}-\frac{L_1\dot{V_{in}}}{R_1}+{V_C}+C_1L_1\ddot{V_C}=0##

I end up with:

##\ddot{V_C}+\frac{\dot{V_C}}{R_1C_1}+\frac{\dot{V_C}}{L_1}+\frac{V_CR_2}{R_1C_1L_1}+\frac{V_C}{C_1L_1}=\frac{V_{in}}{R_1C_1}+\frac{V_{in}}{R_1C_1L_1}##

Circuit #2 (V2=VC)

Node V1

##\frac{V_1-V_{in}}{R_1}+\frac{1}{L}\int_{0}^{t}(V_1)+\frac{V_1-V_2}{R_2}=0##

Node V2

##\frac{V_2-V_1}{R_2}+C\dot{V_2}=0##

Solve NodeV2 for V1

##V_2-V_1+R_2C\dot{V_2}=0##

##V_1=V_2+R_2C\dot{V_2}##

plug into NodeV1

##\frac{V_2+R_2C\dot{V_2}-V_{in}}{R_1}+\frac{1}{L}\int_{0}^{t}(V_2+R_2C\dot{V_2})+\frac{V_2+R_2C\dot{V_2}-V_2}{R_2}=0##

Simplify

##\frac{V_2}{R_1}+\frac{R_2C\dot{V_2}}{R_1}-\frac{V_{in}}{R_1}+\frac{1}{L}\int_{0}^{t}(V_2+R_2C\dot{V_2})+C\dot{V_2}=0##

Derive everything to get rid of integral

##\frac{\dot{V_2}}{R_1}+\frac{R_2C\ddot{V_2}}{R_1}-\frac{\dot{V_{in}}}{R_1}+\frac{V_2}{L}+\frac{R_2C\dot{V_2}}{L}+C\ddot{V_2}=0##

Combine like terms

##C\ddot{V_2}+\frac{R_2C\ddot{V_2}}{R_1}+\frac{\dot{V_2}}{R_1}+\frac{R_2C\dot{V_2}}{L}+\frac{V_2}{L}=\frac{\dot{V_{in}}}{R_1}##

Divide everything by C

##\ddot{V_2}+\frac{R_2\ddot{V_2}}{R_1}+\frac{\dot{V_2}}{CR_1}+\frac{R_2\dot{V_2}}{L}+\frac{V_2}{LC}=\frac{\dot{V_{in}}}{CR_1}##

Simplify

##\ddot{V_2}(1+\frac{R_2}{R_1})+\dot{V_2}(\frac{1}{CR_1}+\frac{R_2}{L})+\frac{V_2}{LC}=\frac{\dot{V_in}}{CR_1}##

Am I doing these two problems right?