Differential Equations Cheatsheet

Cheat Codes & Shortcuts

• Definition: A differential equation relates a function with its derivatives.
• General Form: \( \frac{dy}{dx} = f(x, y) \)
• Order: Highest order of derivative in the equation.
• Separable Equation: \( \frac{dy}{dx} = g(x)h(y) \Rightarrow \int \frac{1}{h(y)} dy = \int g(x) dx \)
• Linear Equation: \( \frac{dy}{dx} + P(x)y = Q(x) \)
• Integrating Factor: \( IF = e^{\int P(x)\,dx} \)
• Homogeneous Equation: \( \frac{dy}{dx} = f\left(\frac{y}{x}\right) \) — use substitution \( y = vx \)
• Exact Equation: \( M(x, y)dx + N(x, y)dy = 0 \) is exact if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \)
• Complementary + Particular: General solution = CF + PI
• 2nd Order Linear: \( a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0 \)

Quick Reference Table

Type	Form	Solution
Separable	\( \frac{dy}{dx} = x e^y \)	\( \int e^{-y} dy = \int x dx \)
Linear (1st Order)	\( \frac{dy}{dx} + y = e^x \)	Use IF \( = e^{\int 1 dx} = e^x \)
Homogeneous	\( \frac{dy}{dx} = \frac{x + y}{x} \)	Substitute \( y = vx \), then solve
Exact	\( (2xy + y^2)dx + (x^2 + 2xy)dy = 0 \)	Check if exact, then integrate \( M \) & \( N \)
Second Order	\( y'' - 3y' + 2y = 0 \)	Auxiliary Eq: \( m^2 - 3m + 2 = 0 \) → \( m=1,2 \)
General Solution	Linear DE	\( y = CF + PI \)

Advice

Differential Equations Quick Tips

• First-Order Linear: \( \frac{dy}{dx} + P(x)y = Q(x) \), use integrating factor \( e^{\int P(x) \, dx} \).
• Separable Equations: Write as \( \frac{dy}{dx} = \frac{g(x)}{h(y)} \), then integrate both sides.
• Homogeneous Equations: Substitute \( y = vx \) to simplify.
• Second-Order Linear: For \( y'' + ay' + by = 0 \), solve characteristic equation \( r^2 + ar + b = 0 \).
• Particular Solution: Use method of undetermined coefficients for non-homogeneous equations.

Differential Equations Speed Quiz

Test your speed with 5 differential equations questions! You have 30 seconds per question.