Differential equations: basics

The essentials

A differential equation is an equation involving an independent variable, a dependent variable and one or more derivatives of the dependent variable. Examples: dy/dx = 2x, and d²y/dx² + y = 0.

The order is the order of the highest derivative present. The degree is the power of that highest-order derivative, but it is defined only when the equation is a polynomial in all its derivatives. If a derivative appears inside sin, log, eˣ and so on, the degree is not defined.

A general solution contains as many arbitrary constants as the order of the equation; a particular solution is obtained by giving those constants definite values, usually from initial conditions such as y(0) = 1.

Key points

• (d²y/dx²)² + (dy/dx)³ + y = 0 has order 2 and degree 2.
• d³y/dx³ + 2 d²y/dx² + dy/dx = 0 has order 3 and degree 1.
• y″ + sin(y′) = 0 has order 2 and its degree is not defined.
• The order and degree (when defined) are always positive integers.
• y = Aeˣ + Be⁻ˣ (two constants) is the general solution of y″ − y = 0, a second-order equation.

Exam tips

• Before reading off the degree, clear fractional powers of derivatives (for example by squaring) so the equation becomes a polynomial in the derivatives.
• To check a proposed solution, differentiate it and substitute into the equation.

Common mistakes

• Taking the degree as the highest power of ANY derivative instead of the highest-order derivative.
• Giving a degree for an equation like y″ + sin(y′) = 0.