Substituting w=y' in the Differential Equation

In summary, the conversation discusses solving a differential equation using a substitution and following a specific procedure. The solution involves finding the homogeneous and particular solutions and using them to find the values of the variables.
  • #1
Turion
145
2
[tex]y'''-5y''+6y'=8+2sinx[/tex]

If I let w=y', would I be able to solve this differential equation? I'm currently stuck and I just want to know if making this substitution is why I am stuck.
 
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  • #2
Have you followed the procedure for solving a linear ODE?

First, solve the homogeneous equation.
Second, find the particular solution.
Third, add the particular solution to the complementary solution (of the homogeneous equation).
 
  • #3
Turion said:
[tex]y'''-5y''+6y'=8+2sinx[/tex]

If I let w=y', would I be able to solve this differential equation? I'm currently stuck and I just want to know if making this substitution is why I am stuck.
Yes, that works nicely.
If we look at the homogenous system, you get the characteristic polynomial, with e^rx as trial solution:
r^2-5r+6=0, giving r=3 and r=2 as possibles. Furthermore, you have w_p1=4/3, for the constant particular solution.
Now, in order to solve for the particular solution 2sinx, you generally will need w_p=Asin(x)+Bcos(x)

Inserting this gives you the two equations in A and B

sin(x): -A+5B+6A=2 goes to: 5A+5B=2
cos(x):-B-5A+6B=0 goes to: 5B-5A=0
Thus, A=B=1/5.
 

1. What is "Substituting w=y' in the Differential Equation"?

"Substituting w=y' in the Differential Equation" is a mathematical technique used to simplify and solve certain types of differential equations. It involves replacing the derivative of a function, y', with a new variable, w, to create a new equation that is easier to solve.

2. Why is it necessary to substitute w=y' in some differential equations?

Substituting w=y' can help to transform a complex differential equation into a simpler form. This can make it easier to solve and provide a better understanding of the behavior of the function.

3. What types of differential equations can be solved by substituting w=y'?

This technique is most commonly used for first-order ordinary differential equations, but it can also be applied to higher-order equations and partial differential equations.

4. How do you substitute w=y' in a differential equation?

To substitute w=y' in a differential equation, you simply replace all instances of y' with w in the original equation. This results in a new equation in terms of w that can be solved using standard methods.

5. What are the advantages of using "Substituting w=y' in the Differential Equation"?

By using this technique, it is often possible to find a solution to a differential equation that would not be possible using other methods. It can also provide a deeper understanding of the function and its behavior.

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