Module 2 Quick Reference Card – Differential Equations

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Separable Equations

dy/dx = g(x)h(y)
dy/h(y) = g(x) dx
Integrate both sides

Steps

Factor right side as g(x)h(y)
Separate variables, integrate
Check for lost equilibria (h(y)=0)

Linear First-Order

y' + P(x)y = Q(x)
μ(x) = e^(∫P dx)
y = (1/μ)∫μQ dx

Steps

Put in standard form y' + Py = Q
Find integrating factor μ
Multiply, recognize (μy)', integrate

Exact Equations

M dx + N dy = 0
Exact if ∂M/∂y = ∂N/∂x
Solution: F(x,y) = C

Steps

Verify ∂M/∂y = ∂N/∂x
F = ∫M dx + g(y)
∂F/∂y = N to find g(y)

Substitution Methods

Bernoulli: y' + Py = Qy^n

Sub v = y^(1-n)

v' + (1-n)Pv = (1-n)Q

Homogeneous: dy/dx = F(y/x)

Sub v = y/x, y = vx

dy/dx = v + x dv/dx

Reduces to separable in v, x

Method Selection Flowchart

Separable? → Linear? → Exact? → Bernoulli? → Homogeneous?

Form	Method	Key Formula
dy/dx = g(x)h(y)	Separation	Integrate both sides
y' + P(x)y = Q(x)	Integrating factor	μ = e^(∫P dx)
M dx + N dy = 0, M_y = N_x	Exact	F(x,y) = C
y' + Py = Qy^n	Bernoulli	v = y^(1-n)
dy/dx = F(y/x)	Homogeneous sub	v = y/x