System of ODE Help

[Mechengstudent]

Hello,

I need assistance with identifying why it is -4x instead of -3x? Please clarify where I have went wrong.

 

[LCKurtz]

It's impossible to edit images. You need to be more careful with your steps. The first mistake is at (1). You haven't properly solved for [MATH]\frac{dy}{dt}[/MATH] so (1) is wrong.

 

[ksdhart2]

I'd begin by solving the two given equations for the derivatives:

\(\displaystyle \frac{dx}{dt} + x + y = 0 \implies \frac{dx}{dt} = -x - y\)

\(\displaystyle \frac{dy}{dt} + 3x - y = \sin(t) \implies \frac{dy}{dt} = \sin(t) - 3x + y\)

Then when you get down to this step, make those substitutions:

\(\displaystyle \frac{d^2x}{dt^2} + {\color{red}\frac{dx}{dt}} + {\color{blue}\frac{dy}{dt}} = 0\)

\(\displaystyle \frac{d^2x}{dt^2} + \left[ {\color{red} -x - y} \right]+ \left[ {\color{blue} \sin(t) - 3x + y} \right] = 0\)

What do you get after collecting like terms and moving the \(\sin(t)\) to the right-hand side?

 

[Mechengstudent]

After collecting like terms, I get:

[MATH]d^2x/dt^2-4x=-sint[/MATH]
ksdhart2, brilliant explanation. I understand now. Thanks!