4.9: Problems

1. Find the first four terms in the Taylor series expansion of the solution to
   a. \(y^{\prime}(x)=y(x)-x, y(0)=2\)
   b. \(y^{\prime}(x)=2 x y(x)-x^{3}, y(0)=1\).
   c. \((1+x) y^{\prime}(x)=p y(x), y(0)=1\).
   d. \(y^{\prime}(x)=\sqrt{x^{2}+y^{2}(x)}, y(0)=1\).
   e. \(y^{\prime \prime}(x)-2 x y^{\prime}(x)+2 y(x)=0, y(0)=1, y^{\prime}(0)=0\).

2. Use the power series method to obtain power series solutions about the given point.
   a. \(y^{\prime}=y-x, y(0)=2, x_{0}=0\).
   b. \((1+x) y^{\prime}(x)=p y(x), x_{0}=0\).
   c. \(y^{\prime \prime}+9 y=0, y(0)=1, y^{\prime}(0)=0, x_{0}=0\).
   d. \(y^{\prime \prime}+2 x^{2} y^{\prime}+x y=0, x_{0}=0\).
   e. \(y^{\prime \prime}-x y^{\prime}+3 y=0, y(0)=2, x_{0}=0\).
   f. \(x y^{\prime \prime}-x y^{\prime}+y=e^{x}, y(0)=1, y^{\prime}(0)=2, x_{0}=0\).
   g. \(x^{2} y^{\prime \prime}-x y^{\prime}+y=0, x_{0}=1\).

3. In Example \(4.3\) we found the general Maclaurin series solution to

\[y^{\prime \prime}-x y^{\prime}-y=0 \nonumber \]

1. Show that one solution of this problem is \(y_{1}(x)=e^{x^{2} / 2}\).
2. Find the first five nonzero terms of the Maclaurin series expansion for \(y_{1}(x)\) and
3. d. Verify that this second solution is consistent with the solution found in Example \(4.3\).

4. Find at least one solution about the \(\operatorname{singular}\) point \(x=0\) using the power series method. Determine the second solution using the method of reduction of order.

a. \(x^{2} y^{\prime \prime}+2 x y^{\prime}-2 y=0\).
b. \(x y^{\prime \prime}+(1-x) y^{\prime}-y=0\).
c. \(x^{2} y^{\prime \prime}-x(1-x) y^{\prime}+y=0\).

5. List the singular points in the finite plane of the following:

1. \(\left(1-x^{2}\right) y^{\prime \prime}+\dfrac{3}{x+2} y^{\prime}+\dfrac{(1-x)^{2}}{x+3} y=0\).
2. \(\dfrac{1}{x} y^{\prime \prime}+\dfrac{3(x-4)}{x+6} y^{\prime}+\dfrac{x^{2}(x-2)}{x-1} y=0\)
3. \(y^{\prime \prime}+x y=0\).
4. \(x^{2}(x-2) y^{\prime \prime}+4(x-2) y^{\prime}+3 y=0\).

6. Sometimes one is interested in solutions for large \(x\). This leads to the concept of the point at infinity.

a. Let \(z=\dfrac{1}{x}\) and \(y(x)=v(z)\). Using the Chain Rule, show that

\[\begin{aligned} \dfrac{d y}{d x} &=-z^{2} \dfrac{d v}{d z} \\ \dfrac{d^{2} y}{d x^{2}} &=z^{4} \dfrac{d^{2} v}{d z^{2}}+2 z^{2} \dfrac{d v}{d z} \end{aligned} \nonumber \]

b. Use the transformation in part (a) to transform the differential equation \(x^{2} y^{\prime \prime}+y=0\) into an equation for \(w(z)\) and classify the point at infinity by determining if \(w=0\) is an ordinary point, a regular singular point, or an irregular singular point.

c. Classify the point at infinity for the following equations:

i. \(y^{\prime \prime}+x y=0\).

ii. \(x^{2}(x-2) y^{\prime \prime}+4(x-2) y^{\prime}+3 y=0\).

7. Find the general solution of the following equations using the Method of Frobenius at \(x=0\).

a. \(4 x y^{\prime \prime}+2 y^{\prime}+y=0\)

b. \(y^{\prime \prime}+\dfrac{1}{4 x^{2}} y=0\)

c. \(x y^{\prime \prime}+2 y^{\prime}+x y=0\).

d. \(y^{\prime \prime}+\dfrac{1}{2 x} y^{\prime}-\dfrac{x+1}{2 x^{2}} y=0\).

d. \(4 x^{2} y^{\prime \prime}+4 x y^{\prime}+\left(4 x^{2}-1\right) y=0\).

e. \(2 x(x+1) y^{\prime \prime}+3(x+1) y^{\prime}-y=0\).

f. \(x^{2} y^{\prime \prime}-x(1+x) y^{\prime}+y=0\).

g. \(x y^{\prime \prime}-(4+x) y^{\prime}+2 y=0\).

8. Find \(P_{4}(x)\) using

a. The Rodrigues Formula in Equation 4.5.1.

b. The three-term recursion formula in Equation 4.5.3.

9. In Equations 4.5.13 through Equation 4.5.20 we provide several identities for Legendre polynomials. Derive the results in Equations 4.5.14 through 4.5.20 as described in the text. Namely,

a. Differentiating Equation 4.5.13 with respect to \(x\), derive Equation 4.5.14.

b. Derive Equation 4.5.15 by differentiating \(g(x, t)\) with respect to \(x\) and rearranging the resulting infinite series.

c. Combining the previous result with Equation 4.5.13, derive Equations 4.5.16 and 4.5.17.

d. Adding and subtracting Equations 4.5.16 and 4.5.17, obtain Equations 4.5.18 and 4.5.19.

e. Derive Equation 4.5.20 using some of the other identities.

10. Use the recursion relation Equation 4.5.3 to evaluate \(\int_{-1}^{1} x P_{n}(x) P_{m}(x) d x, n \leq m\).
11. Consider the Hermite equation

\[y^{\prime \prime}-2 x y^{\prime}+2 n y=0 \nonumber \]

Determine the recursion formula for the coefficients in a series solution, \(y(x)=\sum_{k=0}^{\infty} c_{k} x^{k} .\) Show that if \(n\) is an integer, then one of the linearly independent solutions is a polynomial.

12. Using the power series method to find the general solution of Airy’s equation,

\[y^{\prime \prime}-x y=0 \nonumber \]

13. Use integration by parts to show \(\Gamma(x+1)=x \Gamma(x)\).
14. Prove the double factorial identities:

\[(2 n) ! !=2^{n} n ! \nonumber \]

and

\[(2 n-1) ! !=\dfrac{(2 n) !}{2^{n} n !} \nonumber \]

15. Using the property \(\Gamma(x+1)=x \Gamma(x), x>0\), and \(\Gamma\left(\dfrac{1}{2}\right)=\sqrt{\pi}\), prove that

\[\Gamma\left(n+\dfrac{1}{2}\right)=\dfrac{(2 n-1) ! !}{2^{n}} \sqrt{\pi} \nonumber \]

16. Express the following as Gamma functions. Namely, noting the form \(\Gamma(x+1)=\int_{0}^{\infty} t^{x} e^{-t} d t\) and using an appropriate substitution, each expression can be written in terms of a Gamma function.
    a. \(\int_{0}^{\infty} x^{2 / 3} e^{-x} d x\).
    b. \(\int_{0}^{\infty} x^{5} e^{-x^{2}} d x\)
    c. \(\int_{0}^{1}\left[\ln \left(\dfrac{1}{x}\right)\right]^{n} d x\).

17. A solution of Bessel’s equation, \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-n^{2}\right) y=0\), can be found using the guess \(y(x)=\sum_{j=0}^{\infty} a_{j} x^{j+n} .\) One obtains the recurrence relation \(a_{j}=\dfrac{-1}{j(2 n+j)} a_{j-2}\). Show that for \(a_{0}=\left(n ! 2^{n}\right)^{-1}\), we get the Bessel function of the first kind of order \(n\) from the even values \(j=2 k:\)

\[J_{n}(x)=\sum_{k=0}^{\infty} \dfrac{(-1)^{k}}{k !(n+k) !}\left(\dfrac{x}{2}\right)^{n+2 k} \nonumber \]

18. Use the infinite series in Problem 17 to derive the derivative identities Equation 4.6.4 and 4.6.5:
    a. \(\dfrac{d}{d x}\left[x^{n} J_{n}(x)\right]=x^{n} J_{n-1}(x)\).
    b. \(\dfrac{d}{d x}\left[x^{-n} J_{n}(x)\right]=-x^{-n} J_{n+1}(x)\).

19. Prove the following identities based on those in Problem 18 .

1. \(J_{p-1}(x)+J_{p+1}(x)=\dfrac{2 p}{x} J_{p}(x) .\)
2. \(J_{p-1}(x)-J_{p+1}(x)=2 J_{p}^{\prime}(x) .\)

20. Use the derivative identities of Bessel functions, Equation 4.6.4 and 4.6.5, and integration by parts to show that

\[\int x^{3} J_{0}(x) d x=x^{3} J_{1}(x)-2 x^{2} J_{2}(x)+C \nonumber \]

21. We can rewrite the series solution for Bessel functions,

2. Extend the series definition of the Bessel function of the first kind of order \(v, J_{v}(x)\), for \(v \geq 0\) by writing the series solution for \(y(x)\) in Problem 17 using the Gamma function.
3. Extend the series to \(J_{-v}(x)\), for \(v \geq 0\). Discuss the resulting series and what happens when \(v\) is a positive integer.
4. d. Use the results in part \(c\) with the recursion formula for Bessel functions,

   \[J_{v-1}(x)+J_{v+1}(x)=\dfrac{2 v}{x} J_{v}(x) \nonumber \]

22. Show that setting \(\alpha=1\) and \(\beta=\gamma\) in \({ }_{2} F_{1}(\alpha, \beta ; \gamma ; x)\) leads to the geometric series.

23. Prove the following:
    a. \((a)_{n}=(a)_{n-1}(a+n-1), n=1,2, \ldots, a \neq 0\)
    b. \((a)_{n}=a(a+1)_{n-1}, n=1,2, \ldots, a \neq 0\).

24. Verify the following relations by transforming the hypergeometric equation into the equation satisfied by each function.
    a. \(P_{n}(x)={ }_{2} F_{1}\left(-n, n+1 ; 1 ; \dfrac{1-x}{2}\right)\).
    b. \(\sin ^{-1} x=x_{2} F_{1}\left(\dfrac{1}{2}, \dfrac{1}{2} ; \dfrac{3}{2} ; x^{2}\right)\).
    c. \(J_{p}(x)=\dfrac{1}{\Gamma(p+1)}\left(\dfrac{z}{2}\right)^{p} e^{-i z}{ }_{1} F_{1}\left(\dfrac{1}{2}+p, 1+2 p ; 2 i z\right)\).