Find the general solution of DE 4y "+ y '= 0

y=a+be^(-1/4t)

Step-by-step explanation:

We can solve the characteristic equation, 4r^2+r=0 to help.

Factor:

r(4r+1)=0

This implies r=0 or 4r+1=0.

The second equation can be solved by subtracting 1 on both sides and then dividing boths sides by 4 like so:

4r+1=0

4r+1-1=0-1

4r=-1

4r/4=-1/4

r=-1/4

So r=0 and r=-1/4 which means the general solution is y=ae^(0t)+be^(-1/4t)

Simplifying gives y=a+be^(-1/4t) since e^0=1.

Let's check our solution.

y=a+be^(-1/4t)

y'=0+-1/4be^(-1/4t)

y''=0+1/16be^(-1/4t)

Plug in:

4y''+y'=0

4*1/16be^(-1/4t)+-1/4be^(-1/4t)=0

1/4be^(-1/4t)+-1/4be^(-1/4t)=0

0=0 is a true equation so the solution has been verified.

the central idea is that:

individuals should refuse to cooperate with institutions they believe unjust.

thoreau felt  that governments were more often agents of corruption than justice, even democracies, and that just because the majority feels a certain way does not make it just or correct in absolute terms. he uses the fact that slavery exists in our democracy to support this idea.

he goes on to say that, as a result, people should live true to their own internal laws regarding right and wrong, as a product of reflection and a moral life, rather than following every law that society imposes. 

step-by-step explanation: