Section 4.10  Qualitative Considerations for Variable-Coefficient and Nonlinear Equations

• to be able to use the energy integral lemma to aid in the analysis of solutions to differential equations
• to be able to use the mass-spring oscillator analogy to determine long-term behavior

• energy integral lemma

• mass-spring oscillator analogy

proof of the energy integral lemma

• example: y'' = 3y²
• existence and uniqueness
• linearly independent solutions

 
 
 
 
 
 
 
 
 
 
• analogies to mass-spring oscillations
• Airy equation

 
 
 
 
 
 
 
 
 
 
 
 
• Bessel's equation

 
 
 
 
 
 
 
 
 
 
• Duffing equation

 
 
 
 
 
 
 
 
 
 
 
 
 
• van der Pol equation

 
 
 
 
 
 
 
 
 
 
 
 
 
• pendulum equation