**Answer:**

**Introduction**
In this question, we have two parallel plate capacitors, one with capacitance C and the other with capacitance 2C. These capacitors are connected in parallel and charged to a potential difference V by a battery. Then, the space between the plates of capacitor C is completely filled with a material of dielectric constant K = 3. We need to determine the potential difference across the capacitors after the dielectric is inserted.

**Formula**
The formula for the capacitance of a parallel plate capacitor is given by:

C = (ε₀ * A) / d

Where:
C = Capacitance
ε₀ = Permittivity of free space (constant)
A = Area of the plates
d = Distance between the plates

**Capacitance Calculation**
Let's assume the capacitance of the first capacitor with capacitance C is C₁, and the capacitance of the second capacitor with capacitance 2C is C₂.

C₁ = (ε₀ * A) / d
C₂ = (ε₀ * A) / d

Since the area and distance between the plates are the same for both capacitors, the capacitance of the first capacitor is C₁ = (ε₀ * A) / d, and the capacitance of the second capacitor is C₂ = 2 * (ε₀ * A) / d.

**Equivalent Capacitance**
When capacitors are connected in parallel, the equivalent capacitance is the sum of the individual capacitances.

Ceq = C₁ + C₂
Ceq = (ε₀ * A) / d + 2 * (ε₀ * A) / d
Ceq = 3 * (ε₀ * A) / d

**Effect of Dielectric**
When a dielectric material is inserted between the plates of a capacitor, the capacitance increases by a factor of the dielectric constant (K).

New Capacitance = K * Ceq
New Capacitance = 3K * (ε₀ * A) / d

**Potential Difference Calculation**
The potential difference across the capacitors can be calculated using the formula:

V = Q / C

Where:
V = Potential Difference
Q = Charge
C = Capacitance

Since the capacitors are charged to a potential difference V by a battery, the charge on the capacitors is the same.

V₁ = Q / C₁
V₂ = Q / C₂

After the dielectric is inserted, the new potential difference across the capacitors can be calculated using the new capacitance.

New potential difference across C₁ = Q / (3K * (ε₀ * A) / d)
New potential difference across C₂ = Q / (6K * (ε₀ * A) / d)

Therefore, the potential difference across the capacitors after the dielectric is inserted is V₁ = Q / (3K * (ε₀ * A) / d) and V₂ = Q / (6K * (ε₀ * A) / d).