How to Find Two Linearly Independent Solutions of (y' + f(x)y)' = 0?

In summary, the conversation discusses finding two linearly independent solutions for a second-order differential equation involving a continuous function f on R. It is mentioned that the obvious method of attack, differentiating each term, may not be effective since f may not be differentiable. The derivative of the left-hand-side is noted to be 0, suggesting that the bit in parentheses is a constant. Taking two different values for this constant can lead to two independent solutions to the original equation.
  • #1
shapiro478
7
0
Say f is a continuous function on R. How could I find two linearly independent solutions of (y' + f(x)y)' = 0? Notice that there is no hypothesis about f being differentiable, so the obvious method of attack (taking the derivative of each term in the parenthesis and working off the resultant second-order differential equation) probably isn't a good idea. How does the linearly independent part play into this all?
 
Physics news on Phys.org
  • #2
You know the derivative of the left-hand-side is 0, so the bit in parentheses is a constant. That's probably the place to start.
 
Last edited:
  • #3
Why in the world would the "obvious method of attack" be to differentiate? As dhris said, the integral will be a constant. y' + f(x)y= C where C is an arbitrary constant. Taking two different values for C, say 0 and 1, will give you two different linear, first order, equations to solve for the two independent solutions to the original equation.
 

FAQ: How to Find Two Linearly Independent Solutions of (y' + f(x)y)' = 0?

What does it mean for solutions to be linearly independent?

Linearly independent solutions refer to a set of solutions to a system of linear equations that cannot be expressed as a linear combination of each other. This means that each solution in the set provides unique information and is necessary to fully describe the system.

How do you determine if solutions are linearly independent?

To determine if solutions are linearly independent, you can set up a matrix with the solutions as its columns and perform row reduction. If the resulting matrix has a pivot in each column, the solutions are linearly independent. Alternatively, you can check if the determinant of the matrix formed by the solutions is non-zero.

Why is it important for solutions to be linearly independent?

Linearly independent solutions are important because they form the basis for finding the general solution to a system of linear equations. If solutions are not linearly independent, the general solution cannot be found and the system may have infinitely many solutions or no solutions at all.

Can a system of linear equations have more than one set of linearly independent solutions?

Yes, a system of linear equations can have multiple sets of linearly independent solutions. This is because there may be different combinations of solutions that cannot be expressed as a linear combination of each other but still fully describe the system.

How can linearly independent solutions be useful in real-life applications?

Linearly independent solutions have various applications in fields such as physics, engineering, and economics. For example, in physics, linearly independent solutions can be used to describe the motion of a system and predict future behavior. In economics, they can be used to model relationships between variables and make predictions about market trends.

Similar threads

I Explanation of all "its linearly independent derivatives"
Replies
3
Views
2K
I Number of solutions to the order of the equation
Replies
2
Views
944
I Independent functions as solutions
Replies
2
Views
2K
Wronskian and two solutions being independent
Replies
9
Views
2K
I Solve second order linear differential equation
Replies
7
Views
2K
I Linear independence of functions
Replies
4
Views
1K
I Can I Differentiate Both Sides to Solve y in y^2=ln(1-y^2)?
Replies
2
Views
2K
A Improper boundary in non-linear ODE (pseudospectral methods)
Replies
4
Views
1K
I Finding Max and Min Extremes of a Function with Second Derivatives Equal to Zero
Replies
1
Views
2K
I Understanding Euler Method: Finding Initial Condition of y(0)=1
Replies
7
Views
3K

Hot Threads

Recent Insights

Back
Top