4—Differential Equations 1124.14 Check the algebra in the derivation of then= 0Bessel equation. Explicitly verify that the general expression
fora 2 kin terms ofa 0 is correct, Eq. ( 17 ).
4.15 Work out the Frobenius series solution to the Bessel equation for then=^1 / 2 , s=−^1 / 2 case. Graph both
solutions, this one and Eq. ( 18 ).
4.16 Examine the Frobenius series solution to the Bessel equation for the value ofn= 1. Show that this method
doesn’t yield a second solution for this case either.
4.17 Try using a Frobenius series method ony′′+y/x^3 = 0aroundx= 0.
4.18 Solve by Frobenius seriesx^2 u′′+ 4xu′+ (x^2 + 2)u= 0.
4.19 The harmonic oscillator equation,d^2 y/dx^2 +k^2 y= 0, is easy in terms of the variablex. What is this
equation if you change variables toz = 1/x, getting an equation in such things asd^2 y/dz^2. What sort of
singularity does this equation have atz= 0? And of course, write down the answer fory(z)to see what this sort
of singularity can lead to. Graph it.
4.20 Solve by series solution aboutx= 0: y′′+xy= 0.
Ans: 1 −(x^3 /3!) + (1. 4 x^6 /6!)−(1. 4. 7 x^9 /9!) +···is one.
4.21 From the differential equationd^2 u/dx^2 =−u, finish the derivation forc′in Eq. ( 20 ). Derive identities for
the functionsc(x+y)ands(x+y).
4.22 The chain rule lets you take the derivative of the composition of two functions. The function inverse to
sis the functionf that satisfiesf
(
s(x))
=x. Differentiate this with respect toxand derive thatfsatisfiesdf(x)/dx= 1/
√
1 −x^2. What is the derivative of the function inverse toc?4.23 For the differential equationu′′= +u(note the sign change) use the same boundary conditions for two
independent solutions that I used in Eq. ( 19 ). For this new example evaluatec′ands′. Doesc^2 +s^2 have the
nice property that it did before? What aboutc^2 −s^2? What arec(x+y)ands(x+y)? What is the derivative
of the function inverse tos? toc?