 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, $y^{\prime }=y^2$ contains a square of the unknown function, and $y^{\prime }=y(1-y)$ 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 $y^{\prime }=y^2$, $y(0)=y_0>0$. Its solution of a differential equation isTaking its derivative gives $y^{\prime }=y_0^2/(1-y_{0}t{)}^2=y^2$, 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 $0\le t\le 0.4$, both are valid solutions of a differential equation, but their sum fails the same ordinary differential equation:Already at $t=0$, $s=3$ and its derivative is $1+4=5$, whereas $y^{\prime }=y^2$ would require $9$. At $t=1/4$, the defect $s^{\prime }-s^2$ 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.

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 MIT's worked counterexample to nonlinear superposition.