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Fundamentals of Differential Equations
Coupled Equations. In analyzing coupled equations ofthe form $$\frac{d y}{d t}=a x+b y$$
$$\frac{d x}{d t}=\alpha x+\beta y$$
where a, b, a, and b 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 $$\frac{d y}{d x}=\frac{a x+b y}{a x+\beta y}$$
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 $$\frac{d y}{d t}=-4 x-y$$
$$\frac{d x}{d t}=2 x-y$$