# Mathematical and Computational Methods in Physics and Engineering (fall 2016)

## Daily outline for Wed, Oct 26, 2016 9:10 AM (1 year ago)

### Laplace solutions of ODEs

assigned today:
• standard-Laplace transforms:

I can use Laplace transforms to solve differential equations.

1. Compare and contrast the Laplace transform and eigensystem approach to second order differential equations with constant coefficients.
2. Any problems from chapter 6 review 29-33.
3. For a spring system with m=1, viscosity=2, and k=10, determine a plot of the trajectory of the system if x(0)=x'(0)=0 and the driving function is 2 from t=0 to t=1, -2 from t=2 to t=3 and zero otherwise. Use a convolution approach.

Due today:

# in class

• Quiz on damped SHO using series solution
• interesting that while a1 was zero, the other odds weren't
• Asked what they knew of Laplace transforms
• should make equations easier
• some took a stab at what the transformation was
• some said to look them up in a table
• had them derive the results for $$\mathcal{L}[f'(t)]$$ and $$\mathcal{L}[f''(t)]$$
$$y(t)=\mathcal{L}^{-1}\left[\frac{R(s)}{a s^2+bs+c}\right]$$