Spring 2005

University of Rhode Island

y'
' + 5 y' + 4 y = 2 - x^{2 }+ sin (x) |

y(0) = 1, y'(0) = -1. |

x ' = x + 2y + 2 |

y ' = -2x + 3y -
e^{2t} . |

3. Using the method of Laplace transform, find the solution of the following initial value problem:

x ' = 4x + 2y |

y ' =-3 x - y, |

y(0) = 1, y'(0) = -1. |

4. Compute the first eight terms of the power series solution Σ a

y'' - x^{2} y' + 4y = 0, |

y(0) = 1, y'(0) =
1. |

Find the recurrence relation satisfied by a_{n} .

5. Compute the first
seven terms of the power series solution Σ a_{n }x^{n} about the
initial point for the following initial value problem:

y'' - x y' + x^{2} y = 0, |

y(0) = 2, y'(0) = 0. |

Find the recurrence relation satisfied by a_{n} .

6. Write the Euler's
recursive formula for the following initial value problem:

y' - x^{2}y + cos(x) y^{2}
= 0, |

y(0) = 1. |

Find the first three points (x_{0}, y_{0}), (x_{0}, y_{0}), (x_{0}, y_{0}) of the
numerical solution. Take h = 0.2.

7. Write the Euler's
recursive formula for the following initial value problem:

x ' = 2 x + y^{2 }+ sin(x) |

y ' = x^{3} + 4 y^{
}+ cos(y), |

x(0) = 0, y(0) =
1. |

Find the first three points (x_{0}, y_{0}), (x_{0}, y_{0}), (x_{0}, y_{0}) of the
numerical solution. Take h = 0.1.