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 My=Nx. 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 r1, r2; solutions are er1t and er2t

• §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 (eα+βi=e&alpha·(cosβ+isinβ)) to find sinusoidal solutions; typically eαt·cos(βt) and eαt·sin(βt)

• §3.4: if r1=r2 (repeated root), solutions are ert and t·ert.
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 (YP); the sum of these (c1y1+c2y2+YP) is a general solution.
Use Method of Undetermined Coefficients to find YP: 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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