Physics 401


Numerical Solutions to

Second Order Differential Equations

1. Show the second-order correction to the Taylor-series formula for the second derivative of a function f(x) is
   
   $$-h^2 {f^{(iv)}_n \over 12}$$
   
   
   where $h$ is the stepsize.

2. Use a Taylor-series method to generate an algorithm for estimating the first derivative of a function. What is the size of the error?

3. Generate a recursion relation equation for the following equations using the Taylor-series formula for the second derivative. State the order of the error for each result.
   $$\begin{eqnarray*} &a.\quad &{d^2y \over dt^2} - 2ty = 5 \\ &b.\quad &{d^2y \ove... ... + t^2 = 0 \\ &c.\quad &{d^2y \over dt^2} + e^{y^2} + t^2y = 10 \end{eqnarray*}$$
   
   
   
   
   

4. Hooke's Law states the force exerted by a spring is proportional to its displacement from its equilibrium position or
   
   $$F=-kx$$
   
   
   where $F$ is the force exerted by the spring, $k$ is the spring constant, and $x$ is the displacement from the equilibrium point (see the discussion on harmonic motion in Halliday, Resnick, and Walker).
   1. Write Hooke's Law as linear, second-order differential equation.

   2. What is the general solution of the equation in the previous part? Find the particular solution for the initial conditions: $x(0) = 0 ~ m$, ${dx \over dt}(0) = 2~m/s$. Assume k/m = 1.0 kg/$s^2$.

   3. Use a Taylor-series method to generate an algorithm for solving the differential equation.