AME 3031 Differential Equation Solution Method Decision Tree

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The following outlines the various solution techniques to solving differential equations based on the classification of the differential equation.

Note: this document is evolving. It will be periodically updated as new material is covered in the course.

Ordinary Differential Equations

• linear
  • 1st order: integrating factors (2.1)
  • 2nd order (chapter 3)
    • constant coefficients (3.1)
      • homogeneous: assume exponential solutions
        • distinct real roots => exponential solutions (3.2)
        • complex conjugate roots => sines and cosines (3.4)
        • repeated roots => reduction of order (3.5)
      • nonhomogeneous
        • g(t) = exponentials, sines or cosines, polynomials with positive exponents or products thereof: use undertermined coefficients (3.6)
        • if 2 homogeneous solutions are known, use variation of parameters (3.7)
    • variable coefficients:
      • homogeneous: ?
      • nonhomogeneous: variation of parameters (3.7) will work; however, two linearly independent homogeneous solutions need to be known.
  • nth order (chapter 4)
• nonlinear
  • 1st order
    • separable (2.2)
    • exact (2.3)
    • not separable or exact: ?
  • nth order: ?

Partial Differential Equations

• The Wave Equation
  • fixed-endpoint boundary conditions
    • zero initial velocity and specified initial displacement: Section 10.7, equations 21 and 22
    • zero initial displacement and a specified initial velocity: Section 10.7, equations 35 and 36
    • specified initial displacement and velocity: add the previous two together, Section 10.7 equation 38
  • other boundary conditions
• The Heat Equation
  • fixed boundary temperatures: Section 10.6 equations 16 and 17
  • insulated ends: Section 10.6, equations 35 and 37
• Lapalce's Equation