The discussion centers on solving the system of linear ordinary differential equations (ODEs) represented by $$\frac{dx}{dt} + \frac{dy}{dt}=a-(b+c+d)y-bx.$$ Participants emphasize that with two unknown functions, \(x(t)\) and \(y(t)\), and only one equation, one can express one variable in terms of the other. The approach discussed involves solving for \(U\) and \(V\) sequentially, where \(U\) and \(V\) are derived from the original equation. The strategy of substituting \(U\) into the equation for \(V\) is highlighted as a potentially effective method to derive a solvable equation.
PREREQUISITESMathematicians, students studying differential equations, and researchers working on systems of linear ODEs will benefit from this discussion.
kalish said:I want to solve for $x$ and $y$ from the equation $$\frac{dx}{dt} + \frac{dy}{dt}=a-(b+c+d)y-bx.$$
What is the best strategy?
chisigma said:You have two unknown function x(*) and y(*) and only one equation. That means that You can find x(*) as function of y(*) and y'(*) or vice versa, not both x(*) and y(*)...
Kind regards
$\chi$ $\sigma$
kalish said:$1.$ I solved for $U$.
$2.$ Then I solved the equation for $V$.
$3.$ Now I am plugging in the expression for $U$ into the equation for $V$ (because it looks easier than plugging in the other way around).
Shouldn't this give me a solvable equation for $V$, thus giving $V$, and then let me plug back into the main equation to get $U$??
Ackbach said:Could you please explain what $U$ and $V$ are? I think it'd also be terrific if you could please provide more context for this problem. How did this problem come to you? Is there any other information you have?