Differential Equations Test #1 Review

• §1.1: Definitions (ordinary, partial, linear, nonlinear, order)
• §1.2: Basics (meaning of "solution," general vs. specific, initial conditions)
• §1.3: Direction Fields (what they mean, how to draw, isoclines, solution curves)
• §1.4: Euler's Method (numerical approximation by algebraically "following" direction field)
  • Start with initial conditions (x₀, y₀); let h = change in x each step
  • New x-value: x_n+1 = x_n + h (because h = Δx)
  • New y-value: y_n+1 = y_n + h · y'_n (because y' ≈ Δy/Δx)
  • Re-calculate derivative at each step; move on until finished.

• §2.2: Separable Equations
  • Any that can be written as h(y) · y' = g(x)
  • "Separate" as h(y) dy = g(x) dx, then integrate both sides and solve for y


• §2.3: Linear Equations
  • Any that can be written as y' + P(x) · y = Q(x)
  • Method of Integrating Factors: multiply by unknown μ(x) to get μy' + μPy = μQ
  • Assume μ' = μP; solve by separation to find μ
  • Then μy' + μPy becomes μy' + μ'y, which is d(μy)/dx
  • Integrate both sides and solve for y if possible


• §2.4: Exact Equations
  • Any that can be written as M(x,y) + N(x,y) · y' = 0 IF there is an F(x,y) s.t. F_x = M and F_y = N
  • To check: find M_y (=F_xy) and N_x (=F_yx); if they're equal, it's an exact equation
  • To find F: integrate M dx and integrate N dy; find c-values that make both the same
  • To solve: F_x + F_y · y' can be replaced by dF/dx; since dF/dx = 0, F must be a constant
  • Set F(x,y) equal to a constant c and (if possible) solve for y

• §3.2: Compartmental Analysis ("tank problems")
  • Typically involves a fluid being mixed with a contaminant
  • Draw a diagram and write expressions for rate in and rate out; dS/dt = rate in – rate out
  • Rate in is usually constant; rate out is proportional to current amount of contaminant
  • Be careful if volume of tank changes, or if rate in is not constant!


• §3.3: Thermodynamics
• Generally: dT/dt = K · (M – T) + H + U
• T = inside temp, M = outside temp, H = incidental heating, U = furnace or air conditioner
• M, H, and U can be functions of time (M is often sinusoidal), or they can be constant
• 1/K is the time constant; the time it takes for T to decay to within 37% (1/e) of M


• §3.4: Newtonian Mechanics
• Usually we know something about acceleration (or force) and want to find position
• Fundamental physics facts: F = m·a, a = dv/dt, v = dx/dt; so a = x'' and v = x'



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