MHB How to Solve for \(x\) and \(y\) in a System of Linear ODEs?

[kalish1]

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]

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?

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

$\chi$ $\sigma$

 

[kalish1]

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

$\chi$ $\sigma$

$1.$ 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]

$1.$ 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 $U$??

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?

 

[kalish1]

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?

Sure. I have provided the full details in the following 3 links:
differential equations - Solution to system of nonlinear ODEs - Mathematics Stack Exchange
differential equations - System of $2$ nonlinear DEs - Mathematics Stack Exchange
differential equations - Solving a system of linear ODEs - Mathematics Stack Exchange

People have largely been unable to help. I hope you can help.