Initial Value Problems

• Most problems in the real world are modeled with differential equations because it is easier to see the relationship in terms of a derivative.
• e.g. Newton's Law: F=Ma, $d^2s/dt^2=a=F/M$ (constant acceleration). 2$^{nd}$ order ordinary differential equation.
  • It is ordinary since it does not involve partial differentials.
  • Second order since the order of the derivative is two.
  • The solution to this equation is a function, $s(t)=(1/2)at^2+v_0t+s_0$.
  • Two arbitrary constants, $v_0$ and $s_0$, the initial values for the velocity and position.
  • The equation for s(t) allows the computation of a numerical value for s, the position of the object, at any value for time, the independent variable, t.
• e.g. Harmonic oscillator problem in mechanics,
• e.g. One-dimensional Schrödinger equation in quantum mechanics,
• e.g. One-dimensional Laplace equation in electromagnetic theory, etc.

• Analytical solutions of these equations are often non-existent or very complicated.
• Numerical solutions are the remedy. In terms of solution technique, we can divide differential equations into three groups:

1. Initial Value Problems: In time-dependent problems, the initial state at time t=0 is given and a solution is searched for later t values. For example, in the following quadratic equation
   $$\frac{d^2y}{dt^2}=f(y,y',t)$$
   two initial conditions must be given at t=0, namely $y(0)$ and $y'(0)$ values. (N$^{th}$ order differential equation $\rightarrow$ N initial conditions).
2. Boundary Value Problems.
3. Eigenvalue (characteristic-value) Problems.