Given that ytc1e6tc2e6t is a solution to the differential eq

Given that y(t)=c1e^6t+c2e^6t is a solution to the differential equation y36y=0, where c1 and c2 are arbitrary constants, find a function y(t) that satisfies the conditions

y\"-36y=0

y(0)=8

limt-->+infinity y(t)=0

y(t)=________?

Solution

We are given general solution to the equation

So we first use the given initial condition and get

y(0)=c1+c2=8

Second condition gives us that as t goes to infinity y goes to 0

But now general solution of y has two components:e^(6t) which goes to infinity as t goes to infinity

And other component e^{-6t} which goes to 0 as t goes to infinity

Hence,c1=0 and c2=8

So,

y(t)=8e^{-6t}