In Class Examples

• Integrating Factor
  
  $$y'+2 y = e^{-x}, y(x) = e^{-x}+C e^{-2 x}.$$
  
  
  
  
  $$y' - 2 x y = x, y(0) = 0\to y(x)=-\frac{2}{2}+C e^{x^2}, C=1/2.$$
  
  
  
  
  $$t y'(t) + 2 y ( t) = 4 t^2, y(1)=2.\to y(t)=t^2+\frac{C}{t^2}, C=1.$$
  
  
  
  
  $$y' - 2 y = 4 -x, {\rm\bf CLICK}.$$
  
  
  
  
  $$y'+2 y = x e^{-2 x}\to y(x) = \frac{x^2}{2}e^{-2 x} + C e^{- 2 x}.$$
  
  
  
  
  $$y'-2 x y = 1, \to y(x) = e^{x^2}\int e^{-x^2}d x + C e^{x^2}.$$
  
  

• Separable Equations
  
  $$y'=\frac{x^2}{1-y^2}\to x=(D+3 y - y^3)^{\frac{1}{3}}.$$
  
  
  
  
  $$y' = -\frac{x}{y}\to x^2+y^2=r^2.$$
  
  
  
  
  $$y' = \frac{3 x^2 + 4 x + 2}{ 2(y-1)}, y(0)=-1\to y(x) = 1 - \sqrt{ x^3+2 x^2 +2 x + 4}.$$
  
  
  
  
  $$y'= \frac{y \cos{x}}{1+ 2 y^2}, y(0)=1\to x = {\rm arcsin} y^2+\ln\vert y\vert-1, C=1.$$
  
  

• Word Problems
• Mixing:

  $\Phi$ liters of water per second, with concentration of salt $C$ enters a container, well mixed mixture of water and salt is leaving container at $\Phi$ liters per second. Volume of a container is $V$. Find an amount of salt in a container as a function of time.

• Second Order Constant coefficients
  
  $$y''-y=0; y(0)=2; y'(0)=-1; \to y(x) = e^x/2+\frac{3}{2}e^{-x}.$$
  
  
  
  
  $$y''+5y'+6y=0, y(0)=2, y'(0)=3; \to y(x) = 9 e^{-2 x} - 7 e^{-3 x}.$$
  
  
  
  
  $$4y''-8y'+3y=0, y(0)=2, y'(0)=1/2, \to y(x)=\frac{5}{2}e^{x/2}-\frac{1}{2}e^{\frac{3}{2}x}.{\rm\bf CLICK}$$
  
  
  
  
  $$y''-4y'+4y=0;y(0)=0;y'(0)=1; y(x) = x e^{2x}.$$
  
  
  
  
  $$y''+9 y =0, \to y(x) = C_1 \cos 3x+ C_2 \sin 3x.$$
  
  
  
  
  $$y''+y'+y=0; y(x) = C_1 e^{-x/2}\cos\frac{\sqrt{3}}{2}x + C_2 e^{-x/2}\sin \frac{\sqrt{3}}{2}x .{\bf {\rm\bf CLICK}}$$