Kenneth Kuttler

662 CHAPTER 33. BOUNDARY VALUE PROBLEMS, FOURIER SERIES

Hint: Do something like this:(p(x)y′

)′ z+(λ 1q(x)+ r (x))yz = 0,(p(x)z′

)′ y+(λ 2q(x)+ r (x))zy = 0.

Now subtract and either use integration by parts or show(p(x)y′

)′ z− (p(x)z′)′ y = ((p(x)y′

)z−(

p(x)z′)

y)′

and then integrate. Use the boundary conditions to show that y′ (a)z(a)−z′ (a)y(a)=0 and y′ (b)z(b)− z′ (b)y(b) = 0. The formula 33.11 is called an orthogonality rela-tion and it makes possible an expansion in terms of certain functions called eigen-functions.

14. Here is a really nice result. Suppose you have y,z are both solutions of the differentialequation (

p(x)y′ (x))′+q(x)y(x) = 0

Show that p(x)W (y,z)(x) =C a constant. Here W (y,z) is the Wronskian.

15. In the above problem, change the variables as follows. Let z(x) = p(x) y′(x)y(x) and

determine the equation which results for z. This kind of equation is called a Riccatiequation. In particular, show that

z′+1

p(x)z2 +q(x) = 0

This kind of equation is like a Bernouli equation with exponent 2, but with anotherfunction added in. For more on this, see [29].

16. Suppose in the equation of Problem 14 you have two solutions u,v whose Wronskianis nonzero so they are independent solutions. Suppose that a,b are consecutive zerosof u and that p(x) > 0 on [a,b]. Show that v has exactly one zero in (a,b). This iscalled the Sturm separation theorem. Hint: Use the result of the above mentionedproblem and argue that v(a) ̸= 0 and that you can assume that v(a)> 0 and that u ispositive on the open interval (a,b).

17. Letting [a,b] = [−π,π] , consider an example of a Sturm-Liouville problem which isof the form

y′′+λy = 0, y(−π) = 0, y(π) = 0.

Show that if λ = n2 and yn (x) = sin(nx) for n a positive integer, then yn is a solutionto this regular Sturm-Liouville problem. In this case, q(x) = 1 and so from Problem13, it must be the case that ∫

−π

sin(nx)sin(mx)dx = 0

if n ̸= m. Show directly using integration by parts that the above equation is true.

66214.15.16.17.CHAPTER 33. BOUNDARY VALUE PROBLEMS, FOURIER SERIESHint: Do something like this:(p(x)y')'z+ (Arg(a) +r @)) y= 0,(p(x)z’)'y+(Aog (x) +r(x)) zy =0.Now subtract and either use integration by parts or show(p(x)y’)'z—(p(x)zZ)'y = (py) z- (p@)z)y)'and then integrate. Use the boundary conditions to show that y’ (a) z(a) —z’ (a) y (a) =0 and y’ (b)z(b) —z (b) y(b) = 0. The formula 33.11 is called an orthogonality rela-tion and it makes possible an expansion in terms of certain functions called eigen-functions.Here is a really nice result. Suppose you have y, z are both solutions of the differentialequation ;(p(x)y'(x)) +a(2)y(x) =0Show that p (x) W (y,z) (x) =C a constant. Here W (y,z) is the Wronskian.In the above problem, change the variables as follows. Let z(x) = p(x) oedetermine the equation which results for z. This kind of equation is called a Riccatiequation. In particular, show thatand1! 2z+—~z +q(x) =0P(x)This kind of equation is like a Bernouli equation with exponent 2, but with anotherfunction added in. For more on this, see [29].Suppose in the equation of Problem 14 you have two solutions u, v whose Wronskianis nonzero so they are independent solutions. Suppose that a,b are consecutive zerosof u and that p(x) > 0 on [a,b]. Show that v has exactly one zero in (a,b). This iscalled the Sturm separation theorem. Hint: Use the result of the above mentionedproblem and argue that v(a) # 0 and that you can assume that v(a) > 0 and that u ispositive on the open interval (a,b).Letting [a,b] = [—2, 2] , consider an example of a Sturm-Liouville problem which isof the formy" +Ay=0, y(—m) =0, y(x) =0.Show that if A =n? and y, (x) = sin (nx) for n a positive integer, then y, is a solutionto this regular Sturm-Liouville problem. In this case, g(x) = 1 and so from Problem13, it must be the case that[ sin (nx) sin (mx) dx = 0—1if n 4 m. Show directly using integration by parts that the above equation is true.