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

[shapiro478]

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?

 

[dhris]

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.

 

[HallsofIvy]

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.