Mathematical Tools for Physics - Department of Physics - University

4—Differential Equations 99

4.56

What is the total impedance left to right in the circuit

R 1

R 2

C 1

L 1 C 2

L 2

?

Ans:R 1 + (1/iωC 2 ) + 1

/[

(1/iωL 1 ) + 1/

(

(1/iωC 1 ) + 1/((1/R 2 ) + (1/iωL 2 ))

)]

4.57 Find a Frobenius series solution aboutx= 0fory′′+ycscx= 0, at least to a few terms.

Ans:x−^12 x^2 + 121 x^3 − 481 x^4 + 1921 x^5 +···

4.58 Find a series solution aboutx= 0forx^2 y′′− 2 ixy′+ (x^2 +i−1)y= 0.

4.59 Just as you can derive the properties of the circular and hyperbolic trigonometric functions from
the differential equations that they satisfy, you can do the same for the exponential. Take the equation

u′=uand consider that solution satisfying the boundary conditionu(0) = 1.

(a) Prove thatusatisfies the identityu(x+y) =u(x)u(y).

(b) Prove that the function inverse touhas the derivative 1 /x.

4.60 Find the asymptotic behavior of the Legendre series for thes= 1case.

4.61 Find Frobenius series solutions forxy′′+y= 0.