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Introduction to ODEs

ODE order and type classifier. Enter an equation to judge order, linearity, homogeneity and the number of independent constants.

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F(x, y, y′, …, y⁽ⁿ⁾) = 0
integer 1–4
0 = homogeneous; 1 = nonhomogeneous

📖 Tutorial｜Introduction to ODEs

1. Definition
An equation containing an unknown function y(x) and its derivatives y′, y″, … is an ordinary differential equation (ODE). The order of the highest derivative is the order of the equation.
2. Symbols
SymbolMeaning
y⁽ⁿ⁾n-th derivative of y
norder of the equation
f(x)right-hand side (forcing term)
C₁…Cₙn arbitrary constants in the general solution
3. How this tool works
  • Order = order of the highest derivative present;
  • If the equation is linear in y and its derivatives, it is linear;
  • RHS = 0 means homogeneous, otherwise nonhomogeneous;
  • The general solution of an n-th order ODE contains n independent arbitrary constants.
4. Steps
  1. Enter the highest order n (1–4);
  2. Enter 0 for homogeneous, 1 for nonhomogeneous;
  3. Click Calculate to see the classification and constant count.
5. Example
Example: y″ + 3y′ + 2y = 0. Highest derivative is the second, linear in y, y′, y″, RHS = 0, so it is a second-order homogeneous linear ODE with 2 arbitrary constants.
6. Pitfalls
Order depends only on the highest derivative, not on how many derivatives appear;
Linearity requires y and its derivatives to appear to the first power only; y² or sin y makes it nonlinear;
Homogeneous refers to the RHS, not to algebraic homogeneity.

❓ FAQ｜Introduction to ODEs

What is the order of an ODE?
It is the order of the highest derivative of the unknown function present. In y″+y=0 the highest is the second derivative, so it is second-order.
How to tell linear from nonlinear?
If the equation is linear in y and its derivatives (no y², y·y′, sin y, etc.), it is linear.
Homogeneous vs nonhomogeneous?
Move all terms with y and its derivatives to the left; if the remaining RHS is 0 it is homogeneous, otherwise nonhomogeneous.
Why n arbitrary constants?
An n-th order equation needs n integrations, each introducing an independent constant.
Does this tool solve the equation?
This tool classifies order and type; use the dedicated solver tools for actual solutions.