- **Differential Equations**
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- - **Getting the General solution**
(*http://mymathforum.com/differential-equations/336936-getting-general-solution.html*)

Getting the General solutionOn question 5 I was told to find a particular solution of the equation. I'm having trouble just getting the general solution. Could anyone show me how to get the general solution? http://i.imgur.com/hTeCUGD.jpg |

if you have a linear, second order differential equation with constant coefficients the general solutions is as follows. Let $a y^{\prime \prime} + b y^{\prime} + c y = 0$ form the polynomial $a x^2 + b x + c$ this is known as the characteristic polynomial Find the roots of this polynomial, $r_1, r_2$ If $r_1 \neq r_2$ then the general solution is then given by $y(x) = \large A e^{r_1 x} + B e^{r_2 x}$ where $A,~B$ are arbitrary constants. if $r_1 = r_2 = r$ then the general solution is given by $y(x) = (A + B x)e^{r x} $ |

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Would it be shameful to say I'm having trouble finding the roots..... It doesn't factors nicely and quadratic formula isn't working for me. |

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if you can't find the roots of a polynomial you really have no business messing with differential equations yet. The quadratic formula isn't working for you? It's working for all of us every day! :D $a=b=c=1$ $ = \dfrac{-1 \pm \sqrt{1 - 4}}{2} =-\dfrac 1 2 \pm i\dfrac{\sqrt{3}}{2}$ |

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Thanks..... Edit: I got this root too BTW..... :( |

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-Dan |

This question has almost nothing to do with the general solution of the homogeneous equation. You are asked to find a single particular solution to the non-homogeneous equation. Looking at the questions, one supposes that you've been looking at the method of undetermined coefficients. The book should give you a solid idea of what solutions you should try. PS: complex roots of the characteristic equation aren't "dirty", they represent an important class of solution to the homogeneous equation. |

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http://i.imgur.com/nlEo4E3.jpg |

It helps to replace $\sin^2\!x$ with the equivalent $(1 - \cos(2x))/2$. |

which book is this? |

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