Solving Inhomogeneous Differential Equations with ODE Particular Solution

In summary, the conversation revolves around solving a differential equation, specifically the equation y''+2y=6x+4. The participants discuss different methods for solving it, including inspection, the method of undetermined coefficients, and finding an annihilator. The speaker initially struggles to find a suitable method, but eventually realizes that inspection works well in this case. They also mention trying 2+3x as a possible solution.
  • #1
sleventh
64
0
Hi all,
I've been staring at this for much to long:

y''+2y=6x+4

what were going over now is homogeneous and inhomogeneous differential equations but i can't seem to think of a method for solving this one other then by inspection. Thank you very much for your help.
 
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  • #2
In this case, inspection works quite nicely. What's wrong with that?

If you don't like that, you can try the method of undetermined coefficients or try to find an annihilator of the right hand side.
 
  • #3
i don't like inspection since i haven't found a function that would fit! haha. but thank you, at least i'll stop my wild goose chance for adaptations on integrating factor or separation
 
  • #4
AH! 2+3x works, silly me.
 

FAQ: Solving Inhomogeneous Differential Equations with ODE Particular Solution

What is a particular solution in an ODE?

A particular solution in an ODE (ordinary differential equation) is a specific solution that satisfies the given differential equation, along with any initial conditions. It is different from the general solution, which includes all possible solutions of the ODE.

How do you find a particular solution in an ODE?

To find a particular solution in an ODE, you can use any of the following methods:

  1. Method of undetermined coefficients: This method involves guessing the form of the particular solution based on the form of the non-homogeneous term in the ODE.
  2. Variation of parameters: This method involves finding a particular solution by varying the arbitrary constants in the general solution.
  3. The annihilator method: This method involves finding a particular solution by multiplying the ODE by a suitable function known as an annihilator.

Why is it important to find a particular solution in an ODE?

Finding a particular solution in an ODE is important because it allows us to find a specific solution that satisfies the given differential equation. It is often necessary in real-world applications to have a particular solution in order to make accurate predictions or calculations.

Can a particular solution be unique in an ODE?

No, a particular solution in an ODE is not always unique. Depending on the number of initial conditions and the form of the ODE, there can be multiple particular solutions that satisfy the given differential equation.

What is the difference between a particular solution and a complementary solution in an ODE?

A particular solution in an ODE is a specific solution that satisfies the given differential equation, while a complementary solution is a general solution that includes all possible solutions of the homogeneous part of the ODE. The sum of these two solutions gives the general solution of the ODE.

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