Differential Equations Cheat Sheet
The core ideas of Differential Equations distilled into a single, scannable reference — perfect for review or quick lookup.
PiqCue — piqcue.com/differential-equations/cheatsheet
Quick Reference
Differential Equation
An equation involving a function and its derivatives.
Separable DE
A DE writable as g(y)dy = f(x)dx.
Slope Field
Graphical representation of dy/dx at each point.
Euler Method
Numerical: y_{n+1} = y_n + h*f(x_n, y_n).
Exponential Growth/Decay
dy/dt = ky => y = y_0*e^{kt}.
Initial Value Problem
DE + initial condition y(x_0)=y_0.
General vs Particular Solution
General has C. Particular fixes C via IC.
Key Terms at a Glance
Differential Equation:An equation relating a function to one or more of its derivatives.
Separable DE:A DE that can be written with all y-terms on one side and all x-terms on the other.
Slope Field:A grid of short line segments showing the slope dy/dx at each point.
Euler Method:A first-order numerical method for approximating solutions to DEs.
Initial Value Problem:A DE paired with an initial condition y(x0)=y0.
General Solution:The family of all solutions, containing an arbitrary constant C.
Particular Solution:A single solution determined by applying an initial condition.
Equilibrium Solution:A constant solution where dy/dx=0 everywhere.
Exponential Growth:dy/dt=ky with k>0. Solution grows without bound.
Exponential Decay:dy/dt=ky with k<0. Solution approaches zero.
Half-Life:Time for quantity to reduce to half. t=ln(2)/|k|.
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