Differential Equations Midterm #1 Review
## Differential Equations Midterm #1 Review

__Chapter 1: Basic Concepts__
- §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)

__Chapter 2: First Order Equations__
- §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

__Chapter 3: Second Order Equations__
- §3.1: homogeneous equations with constant coefficients: set up characteristic quadratic and find roots r
_{1}, r_{2}; 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 r
_{1} and r_{2} are complex, use Euler's Formula (e^{α+βi}=e^{&alpha}·(cosβ+*i*sinβ)) to find sinusoidal solutions; typically e^{αt}·cos(βt) and e^{αt}·sin(βt)

- §3.4: if r
_{1}=r_{2} (repeated root), solutions are e^{rt} and t·e^{rt}.

Reduction of order: if one solution y_{1}(t) is known, assume second solution is v(t)·y_{1}(t) and solve original equation as a first order equation for v'(t).

- §3.5: nonhomogeneous equations: find general solution of corresponding homogeneous equation (c
_{1}y_{1}+c_{2}y_{2}) and one particular solution of nonhomogeneous equation (Y_{P}); the sum of these (c_{1}y_{1}+c_{2}y_{2}+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=u
_{1}(t)·y_{1}(t)+u_{2}(t)·y_{2}(t), assume u_{1}'·y_{1}+u_{2}'·y_{2}=0, solve system for u_{1}', u_{2}'.

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