Identification Flowchart:

1. Can variables be separated? => Integrate directly
2. Is it dy/dx = g$\frac{y}{x}$? => Homogeneous, use y = vx
3. Is it dy/dx + Py = Q (linear in y)? => Find IF = e^(integral P dx)
4. Is it dy/dx + Py = $Qy^n$? => Bernoulli, substitute v = y^(1-n)
5. Is M dx + N dy = 0 with dM/dy = $\frac{dN}{dx}$? => Exact, find F
6. Can you spot d(xy), d$\frac{y}{x}$, etc.? => Exact differential shortcut

Must-Remember:

• Order = highest derivative, Degree = power of highest derivative (polynomial only)
• n constants => order n DE
• IF = e^(integral P dx), always set integration constant to 0
• Bernoulli: v = y^(1-n) converts to linear
• Orthogonal trajectories: replace dy/dx with -dx/dy
• Half-life = ln$\frac{2}{k}$
• d(xy) = xdy + ydx, d$\frac{y}{x}$ = $\frac{xdy - ydx}{x}$^2

Common Errors:

• Not dividing by coefficient of dy/dx before finding IF
• Forgetting to substitute back v = $\frac{y}{x}$ after homogeneous solution
• Declaring degree when equation has non-polynomial derivative terms
• Missing the constant of integration in general solutions