Nonlinear Ordinary Differential Equations

Question

Does anyone knows if the following equation $$ x'=\frac{t}{x^4}$$ is a nonlinear ordinary differential equation ?

Because usually the nonlinear ODE is of the form $$x'=tx^4$$

Thanks in advance!

Answer 1

A first-order linear equation has the form

$$ x'(t) + p(t)x(t) = q(t) $$

Your equation does not have this form, so it is non-linear.

If you're confused by the negative power: It doesn't matter, the equation is linear only if it's a linear combination of $x'$ and $x$. For example, all of these equations are non-linear

$$ (x')^2 + \frac{1}{x} =1 $$ $$ x' + \sin(x) = t^3 $$ $$ \frac{1}{\sqrt{x'}} + \ln(t)e^x = 0 $$

Finally, to address the comments: The changing of variables is irrelevant here. Yes, you can change variables to make a new linear equation, but the original equation in $x(t)$ is still very much non-linear.

Answer 2

An equation is linear when the sum of two solutions is also a solution (to a factor $2$), and non linear otherwise.

$$p(t)x'+q(t)x=r(t)$$ is indeed linear, because

$$p(t)(x_1'+x_2')+q(t)(x_1+x_2)=(p(t)x_1'+q(t)x_1)+(p(t)x_2'+q(t)x_2)=r(t)+r(t).$$

You can check that this doesn't work with

$$x'=\frac t{x^4}.$$

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Anyway, by setting $y:=x^5$, you can rewrite as

$$y'=5t,$$ (provided $x\ne0$) which is linear.