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Applied Mathematics · Quick Reference
Differential Equations Cheat Sheet
In one line: a differential equation relates a function to its own rate of change — instead of solving for a number, you solve for an entire function that describes how a system evolves.
Key Ideas
1What is a Differential Equation?. An equation involving a function and one or more of its derivatives, such as dy/dx = 3y.
2Order of a Differential Equation. Determined by the highest derivative appearing in the equation; dy/dx=3y is first-order, while equations with y'' are second-order.
3Separable Differential Equations. Equations that can be rearranged so all y-terms are on one side and all x-terms on the other, then integrated separately.
4Exponential Growth/Decay Model. The equation dy/dt = k*y has the solution y = y0*e^(kt), where y0 is the initial value and k is the growth (or decay, if negative) rate.
5Initial Conditions. A specific starting value (like y(0)=5) needed to pin down one exact solution from the general family of solutions to a differential equation.
Worked Examples
Solve dy/dt = 3y with initial condition y(0)=10.
y = 10*e^(3t)
A radioactive sample decays according to dy/dt = -0.5*y, with y(0)=100. Find y at t=2.
y is approximately 36.8
Classify the order of the differential equation d^2y/dx^2 + 3*dy/dx = 0.
Second-order
Formulas
dy/dt = k*y has solution y = y0 * e^(kt)
Practice Yourself
Solve dy/dt=2y with y(0)=5.
y=5*e^(2t)
What is the order of dy/dx + y = 0?
First-order
What is the order of d^2y/dx^2 = -y?
Second-order