partially decoupled system

[jonl]

the problem given to solve is
dx/dt=3x
dy/dt=y+2x+1
(x(0),y(0))=(3,-1)

I have found x(t)=3e^(3t)
I then attacked y(t) and am stuck after substituting in and getting dy/dt=y+6e^(3t)+1

any nudge in the right direction is appreciated, thank you for your time.

 

[Deleted member 4993]

the problem given to solve is
dx/dt=3x
dy/dt=y+2x+1
(x(0),y(0))=(3,-1)

I have found x(t)=3e^(3t)
I then attacked y(t) and am stuck after substituting in and getting dy/dt=y+6e^(3t)+1

any nudge in the right direction is appreciated, thank you for your time.

dy/dt - y = 6e[sup:bldhdn8m]3t[/sup:bldhdn8m] + 1

Have you done Differential equations yet?

 

[jonl]

yes and i figured out how to do it with substituting and taking the integral, thank you for the time, I am just kicking myself over how much I have forgotten.

 

[Deleted member 4993]

Have you done Differential equations yet?

dy/dt - y = 6e[sup:2wjselhx]3t[/sup:2wjselhx] + 1

This one requires multiplication by "integrating" factor.

For this problem the integrating factor is e[sup:2wjselhx]-t[/sup:2wjselhx]

then multiplying by the intgrating factor, we get,

e[sup:2wjselhx]-t[/sup:2wjselhx] * [dy/dt - y] = e[sup:2wjselhx]-t[/sup:2wjselhx] * [6e[sup:2wjselhx]3t[/sup:2wjselhx] + 1]

d/dt[e[sup:2wjselhx]-t[/sup:2wjselhx] * y] = 6e[sup:2wjselhx]2t[/sup:2wjselhx] + e[sup:2wjselhx]-t[/sup:2wjselhx]

Integrating both sides with respect to 't' - we get:

e[sup:2wjselhx]-t[/sup:2wjselhx] * y = 3e[sup:2wjselhx]2t[/sup:2wjselhx] - e[sup:2wjselhx]-t[/sup:2wjselhx] + C

and continue...