# Ordinary Differential Equations

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The best source of knowledge for ODEs: video lectures from MIT.

I recommend this book: Differential Equations Problem Solvers

## First order, linear, homogeneous equations

1.1 solution Solve the system of ODE's: ${\begin{cases}x_{t}=6x-3y\\y_{t}=2x+y\end{cases}}\,$ 1.2 solution Solve $u'+3u=0,\,\,u(0)=2\,$ 1.3 solution Solve $y'={\frac {y+x}{x}},\,\,\,y(1)=7\,$ ## Second order, linear, homogeneous equations

2.1. solution $y''+3y'+y=0\,$ 2.2. solution $2y''+y'+4y=0\,$ 2.3. solution $y''+4y'+3y=0,\,\,\,y(0)=1,\,y'(0)=0\,$ 2.4. solution $y''+4y'+5y=0,\,\,\,y(0)=1,\,y'(0)=0\,$ ## Second order, linear, nonhomogeneous equations

4.1 solution $y''+2y'+5y=e^{-x}\sin(2x)\,$ 4.2 solution $y''-3y'+2y=3e^{x}\,$ 4.3 solution Find a particular solution of $y''-y'+2y=10e^{-x}\sin(x)\,$ 4.4 solution A ball is thrown straight up from the ground. How high will it go?

## Nonlinear

5.1 solution $x^{2}y''+7xy'+8y=0\,$ 5.2 solution $y'+xy=xy^{2}\,$ 5.3 solution $xy''-y'=3x^{2}\,$ ## Laplace Transforms

The best introduction to the Laplace transform is from an undergrad MIT ODE class, starting at lecture 19 at the link above.

Here are some rules and transforms: http://en.wikipedia.org/wiki/Laplace_transform

6.1 solution $y''-y=e^{-t},\,\,\,y(0)=1,y'(0)=0\,$ 6.2 solution Find the Laplace transform of $e^{at}\,$ 6.3 solution Find the Laplace transform of $f(t)={\begin{cases}1&0 