Question

Determine whether each first-order differential equation is separable, linear, both, or neither. 1. ????y????x+????xy=x2y2 2. y+sinx=x3y′ 3. lnx−x2y=xy′ 4. ????y????x+cosy=tanx

Answer 1

a) Linear

b) Linear

c) Linear

d) Neither

See explanation below.

Explanation:

a)
`(dy)/(dx) +e^x y = x^2 y^2`

For this case the differential equation have the following general form:

`y' +p(x) y = q(x) y^n`

Where
`p(x) =e^x` and
`q(x) = x^2` and since n>1 we can see that is a linear differential equation.

b)
`y + sin x = x^3 y'`

We can rewrite the following equation on this way:

`y' -(1)/(x^3) y= (sin (x))/(x^3)`

For this case the differential equation have the following general form:

`y' +p(x) y = q(x) y^n`

Where
`p(x) =-(1)/(x^3)` and
`q(x) = (sin(x))/(x^3)` and since n=0 we can see that is a linear differential equation.

c)
`ln x -x^2 y =xy'`

For this case we can write the differential equation on this way:

`y' +xy = (ln(x))/(x)`

For this case the differential equation have the following general form:

`y' +p(x) y = q(x) y^n`

Where
`p(x) =x` and
`q(x) = (ln(x))/(x)` and since n=0 we can see that is a linear differential equation.

d)
`(dy)/(dx) + cos y = tan x`

For this case we can't express the differential equation in terms:

`y' +p(x) y = q(x) y^n`

So the is not linear, and since we can separate the variables in order to integrate is not separable. So then the answer for this one is neither.