Question

Write the following ODEs as a system of first-order differential equations. (a) y" + 2y' + 3y = 0 (b) y"' + 2y" + y = 0

Answer 1

a.
`x'_2+2x_2+3 x_1=0`

b.
`x'_2+2x_2+x_1=0`

Explanation:

We are given differential equation of second order in each part

We have to change given differential equation into first order differential equation

a.y''+2y'+3y=0

Suppose
`x_1=y(t)`

`x_2=y'(t)`

Differentiate w.r.t times then we get

`x'_1=y'(t)=x_2`

`x'_2=y''(t)`

Substitute the values in the given differential equation then we get

`x'_2+2x_2+3 x_1=0`

b.y''+2y'+y=0

Suppose

`x_1=y(t)`

`x_2=y'(t)`

Differentiate w.r.t time

Then we get

`x'_1=y'(t)=x_2`

`x'_2=y''(t)`

Substitute the values in given differential equation

`x'_2+2x_2+x_1=0`