Coupled Equations. In analyzing coupled equations of the

Chapter 2, Problem 45E

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QUESTION:

Coupled Equations. In analyzing coupled equations of the form

\(\frac{d y}{d t}=a x+b y\),

\(\frac{d x}{d t}=\alpha x+\beta y\),

where \(a,\ b,\ \alpha\) and \(\beta\) are constants, we may wish to determine the relationship between 𝑥 and 𝑦 rather than the individual solutions \(x(t),\ y(t)\). For this purpose, divide the first equation by the second to obtain

(17)     \(\frac{d y}{d x}=\frac{a x+b y}{\alpha x+\beta y}\).

This new equation is homogeneous, so we can solve it via the substitution \(v=y / x\). We refer to the solutions of (17) as integral curves. Determine the integral

curves for the system

\(\frac{d y}{d t}=-4 x-y\),

\(\frac{d x}{d t}=2 x-y\).

Equation Transcription:

Text Transcription:

{dy}over{dt}=ax+by

{dx}over{dt}=alpha x+beta y

a, b, alpha

beta

x(t), y(t)

{dy}over{dx}={ax+by}over{alpha x+beta y}

v=y/x

{dy}over{dt}=-4 x-y

{dx}over{dt}=2 x-y

Questions & Answers

QUESTION:

Coupled Equations. In analyzing coupled equations of the form

\(\frac{d y}{d t}=a x+b y\),

\(\frac{d x}{d t}=\alpha x+\beta y\),

where \(a,\ b,\ \alpha\) and \(\beta\) are constants, we may wish to determine the relationship between 𝑥 and 𝑦 rather than the individual solutions \(x(t),\ y(t)\). For this purpose, divide the first equation by the second to obtain

(17)     \(\frac{d y}{d x}=\frac{a x+b y}{\alpha x+\beta y}\).

This new equation is homogeneous, so we can solve it via the substitution \(v=y / x\). We refer to the solutions of (17) as integral curves. Determine the integral

curves for the system

\(\frac{d y}{d t}=-4 x-y\),

\(\frac{d x}{d t}=2 x-y\).

Equation Transcription:

Text Transcription:

{dy}over{dt}=ax+by

{dx}over{dt}=alpha x+beta y

a, b, alpha

beta

x(t), y(t)

{dy}over{dx}={ax+by}over{alpha x+beta y}

v=y/x

{dy}over{dt}=-4 x-y

{dx}over{dt}=2 x-y

ANSWER:

Solution

Step 1:

In this problem we have to determine the integral curves of the system.

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