Cannot find nonhomogenous DE solution (should be easy?)

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SUMMARY

The discussion focuses on solving a non-homogeneous differential equation where the user struggles with the substitution method due to an unsolvable system of equations. The key takeaway is the necessity of selecting an appropriate trial function for the particular solution, specifically when the inhomogeneous part shares exponents with the homogeneous solution. The suggested trial function is x_p = a(1, -1)te^{-t} + b(1, -5)e^{-t}, which addresses the overlap in exponents and facilitates finding the correct non-homogeneous solution.

PREREQUISITES
  • Understanding of differential equations, specifically non-homogeneous types.
  • Familiarity with the method of undetermined coefficients.
  • Knowledge of homogeneous solutions and their characteristics.
  • Proficiency in linear algebra concepts, particularly vector notation.
NEXT STEPS
  • Study the method of undetermined coefficients in detail.
  • Learn about the Wronskian and its role in solving differential equations.
  • Explore variations of parameters for non-homogeneous differential equations.
  • Review examples of non-homogeneous differential equations with overlapping exponents.
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Students and educators in mathematics, particularly those focusing on differential equations, as well as anyone seeking to deepen their understanding of solving non-homogeneous systems.

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Homework Statement


Determine the non-homogenous solution of the given differential equation.


Homework Equations


See 3.


The Attempt at a Solution



I have solved for the homogenous part, but as you can see in the link I am getting an unsolvable system of equations with the substitution method on the non-homogenous part. What am I doing wrong?

Capture_zpsc57ed194.jpg

 
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You have to choose an other trial function for xp, as the inhomogeneous part contains the same exponent as one of the exponents in the homogeneous part.

ehild
 
gkirkland said:

Homework Statement


Determine the non-homogenous solution of the given differential equation.


Homework Equations


See 3.


The Attempt at a Solution



I have solved for the homogenous part, but as you can see in the link I am getting an unsolvable system of equations with the substitution method on the non-homogenous part. What am I doing wrong?

Capture_zpsc57ed194.jpg

Try
<br /> x_p = a\begin{pmatrix} 1 \\ -1 \end{pmatrix} te^{-t} <br /> + b\begin{pmatrix}1 \\ -5\end{pmatrix} e^{-t}<br />
 

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