Order and Degree:

• Order = highest derivative order
• Degree = power of highest derivative (equation must be polynomial in derivatives)
• If equation involves sin(y'), e^(y'), etc., degree is not defined

Variable Separable: f(x)dx = g(y)dy => integral f(x)dx = integral g(y)dy + C

Homogeneous DE:

• Test: f(tx, ty) = $t^0$ * f(x, y)
• Substitution: y = vx, dy/dx = v + x$\frac{dv}{dx}$
• Alternative: x = vy if more convenient

Linear First-Order DE:

• Form: dy/dx + P(x)y = Q(x)
• IF = e^(integral P(x) dx)
• Solution: y * IF = integral (Q * IF) dx + C
• If dx/dy + P(y)x = Q(y), treat x as dependent variable

Bernoulli's Equation:

• Form: dy/dx + Py = $Qy^n$ (n != 0, 1)
• Substitute: v = y^(1-n)
• Result: dv/dx + (1-n)Pv = (1-n)Q (linear in v)

Exact DE:

• M dx + N dy = 0 is exact if dM/dy = $\frac{dN}{dx}$
• Solution: F(x,y) = C where dF = M dx + N dy

Common Exact Differentials:

• d(xy) = x dy + y dx
• d$\frac{y}{x}$ = $\frac{x dy - y dx}{x}$^2
• d$\frac{x}{y}$ = $\frac{y dx - x dy}{y}$^2
• d(arctan$\frac{y}{x}$) = $\frac{x dy - y dx}{(x^2+y^2)}$
• d(ln(x^{2+y}^2)) = 2$\frac{x dx + y dy}{(x^2+y^2)}$
• d($e^x$ f(x)) = $e^x$(f(x) + f'(x)) dx

Clairaut's Equation:

• Form: y = xy' + f(y')
• General solution: y = cx + f(c)
• Singular solution: eliminate c from y = cx + f(c) and x + f'(c) = 0

Applications:

• Growth/Decay: dN/dt = kN => N = $N_0$ e^(kt)
• Newton's cooling: dT/dt = -k(T-$T_0$) => T = $T_0$ + ($T_i$ - $T_0$)e^(-kt)
• Orthogonal trajectories: replace dy/dx by -dx/dy