Spring 2005

University of Rhode Island

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

y(0) = 1, y'(0) = -2. |

y' ' - 4 y' + 4 y = e

x^{2
}y''
+ 5 x y' + 29 y = 2 x^{2}
+ x ^{ }^{
} |

y(1) = -2, y'(1) = 1. |

(a) Use the Abel's formula to find a second linearly independent solution of the following differential equation with the given solution:

x^{2
}(ln(x)
- 1) y'' - x y'
+ y = 0 |

y_{1}(x)
= x. |

(b) Find the general solution of the following equation: x

x ' = x + 2y + 2 |

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

6. Find the general solution of the system of differential equations by the matrix method:

x ' = 4x - 2y |

y ' = x + y. |

7. Use the variation of parameters formula to find the particular solution of the following system:

x ' = 4x + 2y + 4 |

y ' = -3 x - y.-3. |