Differential Equations

Differential Equations

1. Partial Differential Equations

   • Is a differential equation that contains unknown multivariable functions and their partial derivatives

2. Ordinary Differential Equations

   • An ordinary differential equation or ODE is an equation containing a function of one independent variable and its derivatives
   • Order
     • The order of a DE is the order of the highest derivative
     • higher order
       • Linear
     • First Order
       • Scalar
         • Linear
           • y'(x) = a(x)y(x)+b(x). For two different solutions y1, y2 the linear combinations ay1+by2 are also solutions.
           • homogenous DE
             • y'(x) = a(x)y(x) so b(x) = 0, and consequently one "automatic" solution is the trivial solution, y = 0
             • Separable Variables: 1. write y' as dy/dx 2. Separate the variables x,y 3. Integrate 4. With the y(x0)=y0 calculate c
             • Substitution: z=y/x or z=px+qy+r
             • y(x) = c * exp(Int( a(t)dt)) from x0 to x
           • inhomogenous DE
             • Formula: y(x) = e^A(x) ( c + Int( b(x) e^-A(x) dx)
             • Variation of parameters: 1. Solution of homogenous problem, get y#. 2. Convert solution to function, c is dependant on t. 3. derive solution y# with respect to t and set it equal to f(x,y(x)) with y# substituting y. 4. From the last equation get the value for c(t). 5. set c(t) in y#. 6. evaluate and get c
         • Non linear
           • Bernoulli type
             • y' (x) = a(x) y + b(x) y^alpha
             • 1. check solutions y=0, x=0 2. set (*) z= y^(1- alpha) 3. solve (*) to y and derivate to y. 4. set (*) = f(x,y(x)) and substitute all y with (*). 5. use Formula or variation of parameters
         • Picard-Lindelöf (First Order): y' (x) = f(t,y(t)) , y(t0) = y0, Let I = [a,b] R a not empty compact interval, x0 el I. Suppose f is Lipschitz cont in y and cont in t in (I x R^n,R^n) . Then there exist a unique solution y(t) C1(I,R^n) to the inital value problem.
           • -An everywhere diff fct g:R->R is Lipschitz cont (with L=sup|g'(x)|) iff it has bounded first derivative. - In particular, any cont diff fct is locally Lipschitz, as cont fct are locally bounded so its gradient its locally bounded as well.
       • Vectorial (constant Coefficients)
         • Linear
           • Homogenous
             • y'(x) = Ay(x)