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It seems you've uploaded an image containing a beam mechanics problem involving a boundary value problem (reuna-arvotehtävä in Finnish) with differential equations and structural analysis.

From the image, the problem involves:
1. A beam subjected to a uniformly distributed load \( q = 2 \, \text{kN/m} \) and a point load \( F = 1 \, \text{kN} \).
2. Boundary conditions with a fixed support on the left and a roller support on the right.
3. The goal is to determine the moment function \( M(z) \) using Galerkin's method.

You are guided through steps to solve the differential equations:
- **Force Equilibrium**: Finding expressions for the shear force \( Q \) and moment \( M \) by solving the differential equations with given boundary conditions.
- **Weak form** of the boundary value problem, requiring integrations and algebraic manipulation to find the moment distribution.

Would you like a detailed step-by-step explanation of how to approach and solve the differential equations for this beam problem? 

Additionally, here are 5 related questions:
1. How does Galerkin's method apply to solving differential equations in structural analysis?
2. What are the key differences between fixed and roller supports in beam problems?
3. How do distributed loads affect the bending moment and shear force along the beam?
4. What are the typical boundary conditions for beams with different types of support?
5. How can finite element methods be used to approximate solutions in beam mechanics?

**Tip:** When working with differential equations in beam theory, always start by identifying the equilibrium conditions and corresponding boundary conditions, which simplify the problem immensely.

Determine the moment function M(z) of a beam subjected to a uniformly distributed load (q = 2 kN/m) and a point load (F = 1 kN) using Galerkin's method and boundary conditions.

Solve for the moment function M(z) of a beam subjected to a distributed load (q = 2 kN/m) and a point load (F = 1 kN). Learn how to apply Galerkin's method to differential equations with boundary conditions and structural analysis.

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