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

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

If you rewrote dy/dx = 3x + 2ydydx=3x+2y as
dy/dx - 2y = 3xdydx2y=3x,
wouldn't your P(x)P(x) be -22 and your Q(x)Q(x) be 3x3x, making this a linear differential equation?

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

2 Answers
Jun 26, 2018

That is linear

Explanation:

With y = y(x)y=y(x), the generalised linear DE is of form:

a_o(x)color(red)(y)+a_1(x)color(red)(y')+a_{2}(x)color(red)(y'')+.... +a_n(x)color(red)(y^((n)))=b(x)

So what you say is true.

bb2 y +(bb(-1)) y' =- 3x

Or, re-arranging:

y' - 2y = 3x

Jun 27, 2018

By definition, a DE is linear when, if y_1 and y_2 are solution of the homogeneous equation, then also any linear combination: y = c_1y_1+c_2y_2 is also a solution of the homogeneous equation.

Given the equation:

dy/dx =3x+2y

the corresponding homogeneous equation is:

dy/dx -2y = 0

Suppose y_1 and y_2 are solutions of the equation, then for any c_1,c_2 let:

y= c_1y_1+c_2y_2

Then:

dy/dx -2y = c_1(dy_1)/dx +c_2(dy_2)/dx -2(c_1y_1+c_2y_2)

dy/dx -2y = c_1((dy_1)/dx-2y_1) +c_2((dy_2)/dx -2y_2)

and as y_1,y_2 are solutions:

dy/dx -2y = c_1*0+c_2*0 = 0

which proves the point.

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

L(f) = 3x

where L(dot) is the differential operator:

f |-> (df)/dx -2f

and the operator L(dot) is linear as:

L(alphaf+betag) = alpha L(f)+ betaL(g)