Test 2: 2nd order ODEs

Test 2: 2nd order ODEs

1. Constant coefficient ODEs

   • Homogeneous: ay''+by'+cy=0
     • Characteristic equation: am^2+bm+c=0
     • General solution: y(x)= c1y1(x)+c2y2(x)
       • y1(x) and y2(x) satisfy y(x)
       • y1(x) and y2(x) are independent
     • solutions form: y(x)=e^(mx)
   • Non-Homogeneous: ay''+by'+cy=h(x)
     • Method of undetermined coefficients
       • General solution: y(x)= c1y1(x)+c2y2(x)+Yp
         • Yp is related to the form of h(x)
         • Characteristic equation: am^2+bm+c=0
         • solutions form: y(x)=e^(mx)
     • Variation of Parameters
       • Green's equation

2. Non-constant coefficients

   • Non-Homogeneous
     • finding solution via Laplace Transform
       • step function
   • Homogeneous
     • Cauchy-Euler equation: h(x)=0
       • two real roots
       • double root
       • imaginary roots
     • Computing series solutions about ordinary points
       • solution: Tylor series

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