Differential Equations Midterm #1 Review

• §1.1: simple models (population, falling); directional fields; equilibrium solutions; end behavior


• §1.2: solving simple differential equations by integration; specific solutions from initial conditions


• §1.3: terminology (order, partial vs. ordinary, linear vs. nonlinear)

• §2.1: integrating factors: works on any y'+p(t)·y=g(t). Multiply by unknown μ(t); assume p·μ=μ', solve for μ, use product rule for μ·y, solve for y.


• §2.2: separable equations: works on any equation that can be written as M(t)=N(y)·y'; useful for some nonlinear cases.


• §2.3: applications (water with contaminant, population growth, heating/cooling). Determine relevant quantities, assign variables, use ideas from physics etc. to establish a differential equation.


• §2.6: exact equations: any equation M(x,y)+N(x,y)·y'=0 in which M_y=N_x. Find ψ(x,y) such that ∂ψ/∂x=M and ∂ψ/∂y=N; then M+N·y'=dψ/dx=0, so ψ=c.


• §2.7: Euler's Method for numerically approximating solutions (algebraically follow the directional field); programming can be useful here

• §3.1: homogeneous equations with constant coefficients: set up characteristic quadratic and find roots r₁, r₂; solutions are e^r₁t and e^r₂t


• §3.2: general theory; any linear combination of solutions is also a solution; the Wronskian determinant; fundamental set of solutions; general solution; Abel's Theorem


• §3.3: if r₁ and r₂ are complex, use Euler's Formula (e^α+βi=e^&alpha·(cosβ+isinβ)) to find sinusoidal solutions; typically e^αt·cos(βt) and e^αt·sin(βt)


• §3.4: if r₁=r₂ (repeated root), solutions are e^rt and t·e^rt.
  Reduction of order: if one solution y₁(t) is known, assume second solution is v(t)·y₁(t) and solve original equation as a first order equation for v'(t).


• §3.5: nonhomogeneous equations: find general solution of corresponding homogeneous equation (c₁y₁+c₂y₂) and one particular solution of nonhomogeneous equation (Y_P); the sum of these (c₁y₁+c₂y₂+Y_P) is a general solution.
  Use Method of Undetermined Coefficients to find Y_P: assume it's some constant A times some function similar to g(t) (see table on pg. 181), then solve for A.


• §3.6: if g(t) is too weird for Undetermined Coefficients, try Variation of Parameters: assume that y=u₁(t)·y₁(t)+u₂(t)·y₂(t), assume u₁'·y₁+u₂'·y₂=0, solve system for u₁', u₂'.

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