1). It is shown how to inverse a matrix using LU decomposition. The LU decomposition is done using Naive Gauss elimination. |

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2). The heat equation at different initial and boundary conditions is solved analytically. To solve the equation is used separation of variables method. |

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3). The analytical solution of the 1D wave equation is given at different initial and boundary conditions. It is used the method of separation of the variable. |

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4). A prey-predator model where the prey has an immature and mature stage structure is proposed and analyzed. Into the phase plane, the existence of attracting is analyzed and defined. The region of equilibrium is defined. The results of the solution can be used for insect biological control. |

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5). Find a Cholesky decomposition of a positive-definite matrix by hand. Solve the linear system of equations using LU decomposition. |

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6). Here is derived the Taylor series for logarithmic function. The relative error is calculated too. |

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7). Pulse response solutions. The convolution between a function and a pulse is derived analytically into the 1D space. |

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8). The 1D diffusion equation at particular initial and boundary conditions is solved analytically. The method of separation of the variable is applied. |

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9). It is applied Naive Gauss elimination to solve a system of three equations. It is shown the Gaussian elimination with partial pivoting too. |

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10). The several problems in fluid dynamics and diffusion are solved using different technics. It is presented two main methods: the method of characteristics and the method of separation of the variable. The Cauchy-Euler, 1D Navier-Stokes, and 2D Laplace equations are solved analytically. |

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