# 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: ${\displaystyle {\begin{cases}x_{t}=6x-3y\\y_{t}=2x+y\end{cases}}\,}$

1.2 solution Solve ${\displaystyle u'+3u=0,\,\,u(0)=2\,}$

1.3 solution Solve ${\displaystyle y'={\frac {y+x}{x}},\,\,\,y(1)=7\,}$

## Second order, linear, homogeneous equations

2.1. solution ${\displaystyle y''+3y'+y=0\,}$

2.2. solution ${\displaystyle 2y''+y'+4y=0\,}$

2.3. solution ${\displaystyle y''+4y'+3y=0,\,\,\,y(0)=1,\,y'(0)=0\,}$

2.4. solution ${\displaystyle y''+4y'+5y=0,\,\,\,y(0)=1,\,y'(0)=0\,}$

## Second order, linear, nonhomogeneous equations

4.1 solution ${\displaystyle y''+2y'+5y=e^{-x}\sin(2x)\,}$

4.2 solution ${\displaystyle y''-3y'+2y=3e^{x}\,}$

4.3 solution Find a particular solution of ${\displaystyle y''-y'+2y=10e^{-x}\sin(x)\,}$

4.4 solution A ball is thrown straight up from the ground. How high will it go?

## Nonlinear

5.1 solution ${\displaystyle x^{2}y''+7xy'+8y=0\,}$

5.2 solution ${\displaystyle y'+xy=xy^{2}\,}$

5.3 solution ${\displaystyle xy''-y'=3x^{2}\,}$

## 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 ${\displaystyle y''-y=e^{-t},\,\,\,y(0)=1,y'(0)=0\,}$

6.2 solution Find the Laplace transform of ${\displaystyle e^{at}\,}$

6.3 solution Find the Laplace transform of ${\displaystyle f(t)={\begin{cases}1&0