To apply mesh analysis, we need to identify meshes. A mesh is a loop that does not contain any other loops within it. In the given circuit, we have two meshes, as shown below:

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Next, we assign currents to each mesh. We will assign clockwise currents to Mesh 1 and Mesh 2, as shown below:

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We will write equations for each mesh using Kirchhoff's voltage law (KVL).

Mesh 1:
-1V + 1Ω(i1-i2) + 2Ω(i1) = 0

Mesh 2:
2Ω(i2-i1) + 3Ω(i2) = 0


We can simplify the equations and solve them using matrix methods or substitution.

After solving the equations, we get:
i1 = 1A
i2 = 0.5A


Now, we can use Ohm's law to calculate the currents in the 1Ω and 2Ω resistances.

Current in 1Ω resistance:
i = (i1 - i2) = (1A - 0.5A) = 0.5A

Current in 2Ω resistance:
i = i2 = 0.5A

Therefore, the current in the 1Ω resistance is 0.5A, and the current in the 2Ω resistance is 0.5A.


In conclusion, using mesh analysis, we were able to find the currents in the 1Ω and 2Ω resistances. The current in the 1Ω resistance is 0.5A, and the current in the 2Ω resistance is 0.5A.