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 largest number of the four options is 1.8 x 10-2

you have it in scientific notation.

1.80 x 10-6   = 0.00000180

1.80 x 10-4   = 0.000180 (bigger, or "less small", than the former one)

so

1.8 x 10-2       = 0.0180. as you see. the "less small", or the biggest one of them all.

as you have negative exponent, it means you are dividing 1.8 by 100 (i.e. moving the dot towards left, two times).

if you had positive exponent for the scientific notation, power of tenths, you would be multiplying times 100 (moving the dot towards right) as below:

1.8 x (10)2 = 180

so. if you have negative exponent in scientific notation, move the dot toward left. if it is positive, toward right as many times the exponent tells you.