TO-DO DIF

Monday review list

1. Fourier series
   • Determine odd/even symmetry
   • Calculate a0,an,bna_0,a_n,b_na0​,an​,bn​
   • Sine and cosine extensions
   • One full integration-by-parts example
2. Heat equation
   • Separation of variables
   • Apply boundary conditions
   • Find μn\mu_nμn​
   • Calculate Fourier coefficients
   • Recognize exponential decay
3. Wave equation
   • Separation of variables
   • Time solution with sine and cosine
   • Use both u(x,0)u(x,0)u(x,0) and ut(x,0)u_t(x,0)ut​(x,0)
   • Calculate AnA_nAn​ and BnB_nBn​
4. Laplace equation
   • One trigonometric direction
   • One hyperbolic/exponential direction
   • Apply four rectangle boundary conditions
   • Match the nonzero boundary using a Fourier series
5. Change of variables for systems

   • Find eigenvalues and eigenvectors

   • Build PPP and DDD

   • Use

     y=P−1xy=P^{-1}xy=P−1x

   • Solve the decoupled equations

   • Transform back with x=Pyx=Pyx=Py

6. Set up a system yourself

   • Define state variables

   • Convert a higher-order equation into first-order equations

   • Write

     x′=Ax+f(t)x'=Ax+f(t)x′=Ax+f(t)

7. CSTR/mixing problem

   • Define amount of salt

   • Use

     rate in−rate out\text{rate in}-\text{rate out}rate in−rate out

   • Track changing or constant volume

   • Solve the resulting linear ODE

8. Complex eigenvalues
   • Find a±bia\pm bia±bi
   • Find a complex eigenvector
   • Extract real and imaginary parts
   • Construct two real solutions
   • Recognize spirals, centers, stability, and rotation direction