= Nonlinear ordinary differential equation
A <nonlinear ordinary differential equation> depends nonlinearly on the unknown <function> or its <derivatives>: it cannot be written in the form defining a <linear differential equation>. For example, $y'=y^2$ contains a square of the unknown <function>, and $y'=y(1-y)$ is a <logistic differential equation>. A <coefficient> such as $t^2$ depending only on the independent variable does not make an <ordinary differential equation> nonlinear. This distinction concerns dependence on the unknown, rather than the shape of its <function graph>.
For a concrete <initial value problem>, consider $y'=y^2$, $y(0)=y_0>0$. Its <solution of a differential equation> is
$$
y(t)=\frac{y_0}{1-y_0t},\qquad -\infty<t<\frac1{y_0}.
$$
Taking its <derivative> gives $y'=y_0^2/(1-y_0t)^2=y^2$, and evaluating at zero checks the <initial condition>. The stated <interval> contains zero and ends at <finite-time blowup>: $y$ tends to positive infinity as $t$ approaches $1/y_0$ from below. Increasing the positive <initial condition> brings <finite-time blowup> closer. This is a simple mathematical example with <dimensionless> $t$ and $y$, rather than a calibrated physical model.
Choose $y_0=1$ and $y_0=2$, giving $y_1=1/(1-t)$ and $y_2=2/(1-2t)$. At $t=1/4$ they have values $4/3$ and $4$, so their sum $s=y_1+y_2$ has value $16/3$. On the common <interval> $0\leq t\leq0.4$, both are valid <solutions of a differential equation>, but their sum fails the same <ordinary differential equation>:
$$
s'-s^2=y_1^2+y_2^2-(y_1+y_2)^2=-2y_1y_2\ne0.
$$
Already at $t=0$, $s=3$ and its <derivative> is $1+4=5$, whereas $y'=y^2$ would require $9$. At $t=1/4$, the defect $s'-s^2$ is $-32/3$. The cross term explains the failure of the <superposition principle>, even though each individual <solution of a differential equation> is a <smooth function>.
\Image[/linear-versus-nonlinear-ode.png]
{title=Two <solutions of a differential equation> for $y'=y^2$ have a sum with a nonzero defect}
The first panel shows $y_1$, $y_2$ and their invalid sum only up to $t=0.4$, before either <finite-time blowup>. The second measures the defect in the <ordinary differential equation>, including the computed value at $t=1/4$. For a <homogeneous linear differential equation>, the corresponding defect of a sum is zero. <Nonlinear ordinary differential equations> can describe state-dependent growth such as the <logistic differential equation>, whose negative quadratic term limits growth instead of accelerating it without bound. Recognizing the distinction tells us when we may assemble <solutions of a differential equation> from simpler pieces and when we must account for their interaction. See https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/be944aa565ebdaa2f9e369e5128d4641_IrRgAWI6bmw.pdf[MIT's worked counterexample to nonlinear superposition].
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