hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

im currently trying to teach myself the D operator technique, as opposed to the 'guess method' which i don't really like.

I stumbled upon this on yahoo answers:

(D^2 + 1)y = 4 cos x - sin x

Find the complementary function by solving the auxiliary equation:

m² + 1 = 0

m² = -1

m = ±i

yᶜ = Asinx + Bcosx

Find the particular integral by comparing coefficients:

yᵖ = Cxsinx + Dxcosx

yᵖ' = (C - Dx)sinx + (Cx + D)cosx

yᵖ'' = (2C - Dx)cosx - (Cx + 2D)sinx

yᵖ'' + yᵖ = 4cosx - sinx

(2C - Dx)cosx - (Cx + 2D)sinx + Cxsinx + Dxcosx = 4cosx - sinx

2Ccosx - 2Dsinx = 4cosx - sinx

2C = 4

C = 2

2D = 1

D = ½

yᵖ = 2xsinx + xcosx / 2

Find the general solution by combining these two parts:

y = yᶜ + yᵖ

y = Asinx + Bcosx + 2xsinx + xcosx / 2

y = (2x + A)sinx + (x + B)cosx / 2

That's all well and good, but from my understanding to solve a DE of the form

y'' -2y' -3y = cos3x

using the D operator technique I would say ok

y_{pi}=1/ (D^{2}-2D-3) cos3x

and then replace D^{2}'s with -(α)^{2}where α is 3 in this case

and then proceed to solve by using D^{2}terms

In the yahoo answers example are they using a different method because there are 2 trig functions on the RHS? I'm also unsure if they have used D as a constant and as the operator?

Sorry for the long post, bottom line is I want to able to use the D operator technique to solve a DE with multiple trig terms on the RHS, and is there a way similar to how you would solve for one trig term, as I explained?

Mitch

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Using the D operator technique to solve a trig based DE

Loading...

Similar Threads for Using operator technique |
---|

I Ode using Fourier Transform |

I Using Complex Numbers to find the solutions (simple Q.) |

I The role of the weight function for adjoint DO |

I Green's function and the evolution operator |

**Physics Forums | Science Articles, Homework Help, Discussion**