Solve Using an Inverse Matrix (dx)/(dt)=y , (dy)/(dt)=-9.8sin(x)-0.5y

Problem

{[d(x)/d(t)=y],[d(y)/d(t)=−9.8*sin(x)−0.5*y])

Solution

1. Linearize the system around the equilibrium point (0,0) by using the small-angle approximation sin(x)≈x

2. Write the linearized system in matrix form d(u)/d(t)=A*u where u=[[x],[y]]

A=[[0,1],[−9.8,−0.5]]

3. Identify the task of solving for specific variables or steady states using an inverse matrix. For a non-homogeneous steady-state system A*u=b the solution is u=A(−1)*b

4. Calculate the determinant of matrix A

det(A)=(0)*(−0.5)−(1)*(−9.8)

det(A)=9.8

5. Find the inverse matrix A(−1) using the formula A(−1)=1/det(A)*[[d,−b],[−c,a]]

A(−1)=1/9.8*[[−0.5,−1],[9.8,0]]

6. Simplify the inverse matrix entries.

A(−1)=[[−0.5/9.8,−1/9.8],[1,0]]

A(−1)≈[[−0.051,−0.102],[1,0]]

Final Answer

A(−1)=[[−5/98,−5/49],[1,0]]

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