# Why isn't #dy/dx = 3x + 2y# a linear differential equation?

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I've learned that linear differential equations are written in the form #dy/dx + P(x)y = Q(x)# .

If you rewrote #dy/dx = 3x + 2y# as

#dy/dx - 2y = 3x# ,

wouldn't your #P(x)# be #-2# and your #Q(x)# be #3x# , making this a linear differential equation?

My teacher told me, however, that this is a nonlinear differential equation.

I've learned that linear differential equations are written in the form

If you rewrote

wouldn't your

My teacher told me, however, that this is a nonlinear differential equation.

##### 2 Answers

That **is** linear

#### Explanation:

With

So what you say is true.

Or, re-arranging:

By definition, a DE is linear when, if

Given the equation:

the corresponding homogeneous equation is:

Suppose

Then:

and as

which proves the point.

You can also look at it in the following way: the equation is in the form:

where

and the operator