Find the solution yt of the following initial value problem

Find the solution y(t) of the following initial value problem, y\" - y = {t, t

Solution

at t<1

y\'\'-y=t

now find general solution

the charactrstic equation of given differential equation is r²-1-0

r=1 and r=-1

the complementry solution is

y(t)=c₁e^t+c₂e^-t

finding general solution

take y(t)=Bt+c

y\'(t)=B

y\"(t)=0

substitute y\'\'-y=t

0-B=t

B=-t

thus general solution is Y=c₁e^t+c₂e^-t-t

now using initial condition

y(0)=0

c₁+c₂=0

y\'(0)=1

c₁-c₂=1

thus on solving

c₂=-1/2

c₁=1/2

thus solutioni s Y=(1/2)e^t-(1/2)e^-t-t

now at t>=1

y\'\'-y=0

the charactrstic equation of given differential equation is r²-1-0

r=1 and r=-1

the complementry solution is

y(t)=c₁e^t+c₂e^-t

finding general solution

using initial condition

y(0)=0

c₁+c₂=0

y\'(0)=1

c₁-c₂=1

thus on solving

c₂=-1/2

c₁=1/2

thus solution is Y=(1/2)e^t-(1/2)e^-t