When is a differential equation nonlinear

An ordinary differential equation or ODE has a discrete finite set of variables. For example in the simple pendulum , there are two variables: angle and angular velocity. A partial differential equation or PDE has an infinite set of variables which correspond to all the positions on a line or a surface or a region of space. For example in the string simulation we have a continuous set of variables along the string corresponding to the displacement of the string at each position.

In practice we approximate the infinite set of variables with a finite set of variables spread across the string or surface or volume at each position.

For an ODE, each variable has a distinct differential equation using "ordinary" derivatives. For a PDE, there is only one "partial" differential equation for each dimension. The order of a differential equation is equal to the highest derivative in the equation.

The single-quote indicates differention. So x ' is a first derivative, while x '' is a second derivative. Linear just means that the variable in an equation appears only with a power of one. This immediately shows that there exists a solution to all first order linear differential equations. This also establishes uniqueness since the derivation shows that all solutions must be of the form above.

This proves that the answers to both of the key questions are affirmative for first order linear differential equations. Notice that the theorem does not apply, since the differential equation is nonlinear. We can separate and solve. How to distinguish linear differential equations from nonlinear ones? I know, that e. You can analyse functions term-by-term to determine if they are linear, if that helps.

The first time a term is non-linear, then the entire equation is non-linear. Makes it much easier. See, I was also overthinking this, but realised you have to go back to those definitions we're given.

The dependent variable y and its derivatives are of first degree; the power of each y is 1. Its graph is a line, i. One could define a linear differential equation as one in which linear combinations of its solutions are also solutions.

Linear Differential equations are those in which the dependent variable and its derivatives appear only in first degree and not multiplied together. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams?

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