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Coupled Equations. In analyzing coupled equations of the
Chapter 2, Problem 45E(choose chapter or problem)
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.