Ch 02/Classification of Differential Equations

Classification of Differential Equations

DEs are classified by type (ODE/PDE), order, degree, and linearity.

IODE vs PDE

An ODE (Ordinary DE) involves derivatives with respect to one independent variable.

A PDE (Partial DE) involves partial derivatives with respect to two or more independent variables.


Your 2nd semester covers ODEs exclusively.

Note

PDEs appear in heat flow, wave propagation, and quantum mechanics. ODEs appear whenever you track one variable over time — growth, decay, oscillation, circuits.

ODE (your focus)
PDE (later semesters)
Example 1easy

Identify whether each equation is an ODE or PDE: (a) (b)

Show step-by-step solution+
Equation (a)
Only one variable x, derivative → ODE
Equation (b)
Two independent variables x and y, partial derivatives → PDE (Laplace equation)

IIOrder and Degree

Order = the order of the highest derivative present.

Degree = the power of the highest-order derivative, after clearing all radicals and fractions from the derivatives.

Note

Always clear fractions and radicals from the derivative terms before reading off the degree. Example: has degree 2, not 1, because squaring gives .

EquationOrderDegreeWhy
11Highest derivative is , raised to power 1
21Highest derivative is , power 1
23Highest derivative raised to power 3
12After squaring:
Example 1easy

Find order and degree of .

Show step-by-step solution+
Identify highest derivative
— this is a 3rd derivative → Order = 3
Find its power
The 3rd derivative appears to the power 1 (no exponent) → Degree = 1
Note
The term does not affect order or degree — only the highest derivative matters.
Example 2medium

Find order and degree of .

Show step-by-step solution+
Identify highest derivative
→ Order = 2
Clear the fractional power
Square both sides:
Read off degree
Now highest derivative is raised to power 2 → Degree = 2

IIILinear vs Non-Linear

A DE is linear when:

- The unknown function y and all its derivatives appear to the first power only

- There are no products between y and any of its derivatives

- The coefficients may be any function of x


Violate any condition → non-linear.

Note

Linearity is crucial because linear DEs have much more powerful solution techniques. The superposition principle holds: if and are solutions, so is .

Linear ✓
Non-linear ✗
Example 1easy

Classify each as linear or non-linear: (a) (b) (c)

Show step-by-step solution+
Equation (a)
, , and all appear to power 1, no products → Linear
Equation (b)
is a product of the function and its derivative → Non-linear
Equation (c)
means raised to power 2 → Non-linear
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Ch 1 · Introduction to Differential Equations
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Ch 3 · Formation of Differential Equations
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