Thread: Is this ODE linear?

If I have the ODE:

x`=x²*e^t+e^x

can this differential equation be considered linear?

I know that the definition of a linear ODE is that it can be written as:
x`+a(t)x=b(t)

if I take the sqrt of my initial ODE than I have:
x'-(e^t)^(1/2)*x=(e^x)^(1/2)

This would be a linear differential equation with:
a(t)=(e^t)^(1/2) and
b(t)=(e^x)^(1/2)

is this true?

Originally Posted by infernalmich

If I have the ODE:

x`=x²*e^t+e^x

can this differential equation be considered linear?

I know that the definition of a linear ODE is that it can be written as:
x`+a(t)x=b(t)

if I take the sqrt of my initial ODE than I have:
x'-(e^t)^(1/2)*x=(e^x)^(1/2)

This would be a linear differential equation with:
a(t)=(e^t)^(1/2) and
b(t)=(e^x)^(1/2)

is this true?

No, it is not true "b(t)= (e^x)^(1/2)" is impossible because the right side is a function of x, not t.

If you thinking, "It is a function of t because x is a function of t", that is true but not the point here. In deciding whether a differential equation is linear or non-linear we treat the variables as if they were independent.

ok, now I understand!

Thank you for your help. But if it would be (e^t)(^1/2) instead of (e^x)(^1/2) it would be linear, right?