All you ever wanted to know about DFQ

1. DFQ classification: ODE/PDE, single/system, First /Second order, Linear/Nonlinear, homogeneous/inhomogeneous
2. Geometric interpretation: $y'(x)=f(x,y(x))$ is the slope of $y(x)$ curve at position $(x,y)$. Plot Slope, connect line segments
3. First Order Linear ODE:
   

   $$y'(x)+ p(x) y(x) = g(x),$$ (21)

   
   
   for given $p(x),g(x)$. To solve, rewrite as
   
   $$\frac{d}{d x}\left( y(x) e^{\int p(x) d x} \right) = g(x) e^{\int p(x) d x},$$
   
   
   which is equivalent to (21), but easy to solve.
4. Separable Equation
   
   $$y'(x)=\frac{M(x)}{N(y)},$$
   
   
   with the implicit solution
   
   $$\int M(x) d x = \int N(y) d y,$$
   
   

5. Radioactive decay, or compound interest
   
   $$\frac{d y(t) } {d t} = \alpha y(t),$$
   
   
   with the solution
   
   $$y(t)=y(t=0)e^{\alpha t},$$
   
   
   for positive or negative $\alpha$.
6. Compound interest with salary, or mixing
   
   $$\frac{d y(t) } {d t} = \alpha y(t) +s, y(0)=y_0$$
   
   
   with the solution
   
   $$y(t) = y_0 e^{\alpha t} + \frac{s}{\alpha}(e^{\alpha t} -1).$$
   
   

7. Logistic Model
   
   $$\frac{d N}{d t} = r (1 - \frac{N}{k})N,$$
   
   
   with the solution
   
   $$N(t) = \frac{k}{1-e^{-r(t+C)}}.$$
   
   

8. Phase Plane To solve
   
   $$y'(x) = f (y(x)),$$
   
   
   plot $y'(x)$ versus $y(x)$.

   Then

   if $y'(x)=0$, then $y(x)={\rm const},$
   if $y'(x)>0$, then $y(x) {\rm increases},$
   if $y'(x)<0$, then $y(x)={\rm decreases}.$

9. Second order Linear ODE:
   
   $$y''(x)+p(x)y'(x)+q(x)y(x)=g(x).$$
   
   

10. Wronskian = $y_1(x)y_2'(x)-y_2(x)y_1'(x).$
11. Theorem: if $W(x)\ne 0$ for solutions $y_1(x)$ and $y_2(x)$, then
    
    $$y(x) = C_1 y_1(x)+C_2 y_2(x),$$
    
    
    is the general solution of the homogeneous second order linear ODE.
12. 
    
    $$e^{i x } = \cos{x}+i \sin{x}.$$
    
    

13. Second order linear homogeneous ODE with constant coefficients
    
    $$a y''(x)+ b y'(x) + c y(x) = 0.$$
    
    
    look for solutions as
    
    $$y(x) = e^{r x},$$
    
    
    and get a quadratic equation for $r$:
    
    $$a r^2 + b r + c = 0, r_{1,2}=\frac{-b\pm \sqrt{b^2 - 4 a c}}{2 a}.$$
    
    
    There are three possibilities:
    1. Both roots are real, and different, then general solution is
       
       $$y(x) = c_1 e^{r_1 x}+ c_2 e^{r_2 x}.$$
       
       

    2. Both roots are real and equal, then the general solution is
       
       $$y(x) = C_1 e^{ r x} + C_2 x e^{r x}.$$
       
       

    3. Both roots are complex, and are complex conjugate of each other:
       
       $$y(x) =C_1 \sin {\Im{r}x}e^{\Re{r}x}+ C_2 \cos{\Im{r}x} e^{\Re{r}x}.$$
       
       

14. Euler Equation Consider equation of the form
    
    $$(x-x0)^2 y''(x) + \alpha (x-x0) y'(x) + \beta y(x) = 0,$$
    
    
    look for a solution in a form
    
    $$y(x) = \vert x- x0\vert^\lambda,$$
    
    
    and obtain quadratic equation for $\lambda$:
    
    $$\lambda(\lambda - 1) + \alpha \lambda + \beta = 0.$$
    
    
    Then there are three cases:
    1. Two real roots which are not equation to each other with the general solution
       
       $$y(x) = C_1 \vert x-x0\vert^{\lambda_1} + C_2 \vert x-x0\vert^{\lambda_2}.$$
       
       

    2. Two real roots which are equal
    3. Two complex conjugate roots
       
       $$\lambda = \mu \pm i \omega,$$
       
       
       with the general solution
       
       $$y(x) = C_1 \vert x-x0\vert^\mu \cos(\lambda\log\vert x-x0\vert) + C_2 \vert x-x0\vert^\mu \sin(\lambda\log\vert x-x0\vert).$$
       
       

15. The concept of Wronskian
16. linear second order ODE with RHS (non homogeneous equations).
    The general solution of the inhomogeneous equation is the general solution of corresponding homogeneous equation plus a special solution of the inhomogeneous equation
17. Method of undetermined coefficients to find inhomogeenous solution
18. Method of variation of a parameter to find inhomogeneous solution
19. Oscillators and oscillations
    1. free oscillator
    2. Damped oscillator (over/critically/under damped oscillators)
    3. Forced oscillator (resonant and nonresonant forcing)
    4. Damped and Forced oscillator. Method of complex amplitude.
20. Fourier Series
    1. Sine Fouries Series
       
       $$A_n = \frac{2}{L}\int\limits_0^L f(x)\sin(\frac{n \pi x}{L}) d x,$$
       
       
       and
       
       $$F(x) = \sum\limits_{n=1}^\infty A_n \sin(\frac{n \pi x}{L}).$$
       
       

    2. Cosine Fouries Series
       
       $$B_n = \frac{2}{L}\int\limits_0^L f(x)\cos(\frac{n \pi x}{L}) d x,$$
       
       
       and
       
       $$F(x) = \frac{B_0}{2}+\sum\limits_{n=1}^\infty B_n \cos(\frac{n \pi x}{L}).$$
       
       

    3. Full Fourier Series
       
       $$A_n = \frac{1}{L}\int\limits_{-L}^L f(x)\sin(\frac{n \pi x}{L}) d x,$$
       
       
       and
       
       $$B_n = \frac{1}{L}\int\limits_{-L}^L f(x)\cos(\frac{n \pi x}{L}) d x,$$
       
       
       and
       
       $$F(x) = \sum\limits_{n=1}^\infty A_n \sin(\frac{n \pi x}{L})+... ...c{B_0}{2}+\sum\limits_{n=1}^\infty B_n \cos(\frac{n \pi x}{L}).$$
       
       

    4. Odd and Even functions. Odd and even extensions. Periodic extension.
21. Boundary Value Problems
22. Heat Equation