Systems of Differential Equations - Examples & Applications

Conceptual Examples: Phase Portraits and Stability

Phase Portrait Classification (Conceptual)

A 2×22 \times 2 linear homogeneous system x′=Ax\mathbf{x}' = A\mathbf{x} has eigenvalues λ1=3\lambda_1 = 3 and λ2=5\lambda_2 = 5. Describe the phase portrait and classify the stability of the critical point at the origin.

Phase Portrait Solution

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Phase Portrait Stability (Complex Roots)

A linear system has the coefficient matrix A=(−12−2−1)A = \begin{pmatrix} -1 & 2 \\ -2 & -1 \end{pmatrix}. Determine the eigenvalues and use them to classify the critical point at the origin.

Complex Stability Solution

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Elimination Method

Elimination Method Problem (Homogeneous)

Solve the system:
x′=2x−yx' = 2x - yy′=xy' = x

Elimination Method Solution

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Elimination Method Problem (Non-homogeneous)

Solve the non-homogeneous system using the elimination method:
x′=x+y+tx' = x + y + ty′=4x+y−ty' = 4x + y - t

Non-homogeneous System Solution

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Matrix Method

Matrix Method (Distinct Real Roots)

Solve the system:
x′=(4233)x\begin{aligned} \mathbf{x}' = \begin{pmatrix} 4 & 2 \\ 3 & 3 \end{pmatrix} \mathbf{x} \end{aligned}

Matrix Method Distinct Solution

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Matrix Method (Complex Roots)

Solve the following system:
x′=(−1−41−1)x\begin{aligned} \mathbf{x}' = \begin{pmatrix} -1 & -4 \\ 1 & -1 \end{pmatrix} \mathbf{x} \end{aligned}

Complex Roots Matrix Solution

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Matrix Method (Repeated Real Roots)

Solve the system:
x′=(3−182−9)x\begin{aligned} \mathbf{x}' = \begin{pmatrix} 3 & -18 \\ 2 & -9 \end{pmatrix} \mathbf{x} \end{aligned}

Repeated Roots Matrix Solution

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