How to Solve and Verify Second Order Inhomogeneous ODEs?

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Discussion Overview

The discussion centers around solving a second-order inhomogeneous ordinary differential equation (ODE) using the complementary function and particular integral method. Participants seek to find the solution that satisfies specific initial conditions and inquire about verifying the solution using technology.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant presents the ODE and requests assistance in finding the solution that meets the initial conditions y(0) = 1 and y'(0) = 0.
  • Another participant provides a proposed solution involving the complementary function and particular integral, including specific values for constants derived from initial conditions.
  • A participant expresses confusion regarding the solution process and requests further clarification on the steps taken.
  • Another participant explains the derivation of the characteristic equation for the homogeneous part and suggests a form for the particular solution based on the right-hand side of the ODE.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the solution process, as there are indications of confusion and requests for clarification. Multiple approaches and interpretations of the solution steps are present.

Contextual Notes

Some participants may be missing specific assumptions or steps in the solution process, and there is a reliance on the correct identification of the complementary and particular solutions.

Who May Find This Useful

Students and individuals interested in solving second-order inhomogeneous ODEs, particularly those seeking to understand the application of the complementary function and particular integral method.

vj9
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Hello All,

I am stuck on the following question. Can you please help to find the solutions

Using the complementary function and particular integral method, find the solution of the diffential equation which satisfies y(0) = 1 and y'(0) = 0.

y'' + 3y' + 2y = 20cos2x

and then can you help about how to check the answer using technology.
 
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D^2 + 3D +2=0
c1e^(-x) +c2e^(-2x) +(3sin2x -cos2x)/10
c1+c2-0.1=1
-c1-2c2 +6=0
c2=4.9, c1=-3.8
 
Can you be more specific . I tried to solve the way u suggested but i am stuck.

Many Thanks,
Vj9
 
First he found the characteristic equation for the homogeneous equation, [itex]D^2+ 3D+ 2= (D+ 1)(D+ 2)= 0[/itex] so that D= -1 and -2 and the "complementary solution" is [itex]C_1e^{-x}+ C_2e^{-2x}[/itex].

Since the right hand side is 20cos(2x), you look for a specific solution of the form Acos(2x)+ B sin(2x). Put that into the equation and solve for A and B.
 

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