Question

It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation -- (what is the highest number of derivatives involved) and whether or not the equation is linear .

Linearity is important because the structure of the the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly. Determine whether or not each equation is linear:
1. (1+y2)(d2y/dt2)+t(dy/dt)+y=et
2. t2(d2y/dt2)+t(dy/dt)+2y=sin t
3. (d3y/dt3)+t(dy/dt)+(cos2(t))y=t3
4. y''-y+y2=0

Answer 1

1. Second-order nonlinear ordinary differential equation.
2. Second-order linear ordinary differential equation.
3. Third-order linear ordinary differential equation.
4. Second-order nonlinear ordinary differential equation

Explanation:

The objective is to determine whether or not each of the following equation is linear:

1. 
   `(1+y^2)(d^2y)/(dt^2) + t (dy)/(dt) = e^t`
2. 
   `t^2(d^2y)/(dt^2) + t (dy)/(dt) +2y= \sin t`
3. 
   `(d^3y)/(dt^3) + t (dy)/(dt) + \cos (2t) y = t^3`
4. 
   `y''-y+y^2 = 0`

`(1)`

We can rewrite this equation in the form

`(1+y^2)y''(t) +ty'(t) = e^t`.

As we can see, this is an second-order nonlinear differential equation, because of the term
`1+y^2` next to
`y''(t)`.

`(2)`

We can rewrite this equation in the form

`t^2y''(t) + t y'(t) +2y= \sin t`.

This is an second-order linear ordinary differential equation.

`(3)`

We can rewrite this equation in the form

`y'''(t)+ t y' + \cos (2t) y = t^3`

This is an third-order linear ordinary differential equation.

`(4)`

This is an second-order nonlinear ordinary differential equation, because of the term
`y^2`.