Answer to Question #213422 in Differential Equations for xiang

Question

given the system express y as a function of t

1. 3dx/dt+2dy/dt-x+y=t-1
2. dx/dt+dy/dt-x=t+2

Answer 1

Solution

Let x’=dx/dt, y’= dy/dt

Solving the system with respect to x 'and y'

x’+x+y=-t-5

y’-2x-y=2t+7

From second equation x=(y’-y-2t-7)/2, x’=(y’’-y’-2)/2

Substituting this into the first equation

(y’’-y’-2)/2+(y’-y-2t-7)/2+y=-t-5

y’’-y’-2+y’-y-2t-7+2y=-2t-10

y’’+y=-1

y(t)=Ce^kt => k²+1=0 => k_1,2=±I => y(t)=Asin(t)+Bcos(t)-1

Answer

y(t)=Asin(t)+Bcos(t)-1