DE-BOOK Numerical Solutions for Linear Systems

Section 14.6 Numerical Solutions for Linear Systems

We’ve explored what linear systems are, how to visualize their behavior, and even how to write them neatly in vector form. When an exact solution is hard—or when we only need an approximation—we can compute solutions numerically.

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We’ll approximate solutions step by step with Euler’s Method and use those steps to see how the variables influence each other.

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Subsection Euler’s Method

Not every system is easy to solve exactly. Sometimes, you just want to see what the solution looks like — or approximate it for a while. That’s where numerical methods shine.

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One of the simplest is Euler’s Method, which you’ve seen for single equations. We’ll now apply it to systems.

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Suppose we have:

\begin{align*} \frac{dx}{dt} \amp = x + y\\ \frac{dy}{dt} \amp = -x + y \end{align*}

with initial conditions \(x(0) = 1\text{,}\) \(y(0) = 0\text{,}\) and a step size of \(h = 0.1\text{.}\)

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🌌 Example 288. Euler’s Method for a System.

Use Euler’s method to take two steps for the system above.

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Solution.

Step 1: At \(t = 0\text{,}\) we have \(x_0 = 1\text{,}\) \(y_0 = 0\text{.}\) The derivatives are \(x' = 1 + 0 = 1\text{,}\) \(y' = -1 + 0 = -1\text{.}\)

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Update:

\begin{equation*} x_1 = 1 + 0.1(1) = 1.1, \qquad y_1 = 0 + 0.1(-1) = -0.1\text{.} \end{equation*}

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Step 2: At \(t = 0.1\text{,}\) we have \(x_1 = 1.1\text{,}\) \(y_1 = -0.1\text{.}\) Now \(x' = 1.1 + (-0.1) = 1.0\text{,}\) \(y' = -1.1 + (-0.1) = -1.2\text{.}\)

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Update again:

\begin{equation*} x_2 = 1.1 + 0.1(1.0) = 1.2, \qquad y_2 = -0.1 + 0.1(-1.2) = -0.22\text{.} \end{equation*}

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We now have approximate values after two steps:

\begin{equation*} (x(0.2), y(0.2)) \approx (1.2, -0.22). \end{equation*}

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Each step shows how \(x\) and \(y\) “talk” to each other over time.

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Euler’s Method may be simple, but it makes the flow of information between variables visible: each update for \(x\) affects \(y\) on the next step, and vice versa.

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Checkpoint 289. 🤔💭 Numerical Solutions for Linear Systems Reading Questions.

(a) Euler’s Method Thinking.

What does Euler’s Method do when applied to a system of DEs? (Select all that apply.)

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It updates each variable step by step using its derivative at the current point.
It shows how the variables influence each other from one step to the next.
It provides the exact solution in one step.
Euler’s Method is an approximation technique, not an exact one-step solution.
It can only be used for uncoupled systems.
Euler’s Method works for any system, coupled or not.

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(b) Interpreting Numerical Results.

When we use Euler’s Method on a system, what do the computed points \((x_n, y_n)\) represent?

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They are approximate values of the solution at specific time steps.
They are exact values that replace the need for solving the system algebraically.
They’re approximate — we still use exact methods when possible.
They are estimates of the system’s coefficients.
Euler’s Method updates \(x\) and \(y\text{,}\) not the coefficients.

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