Chapter 5

Use the power series method to solve the given initial-value problem. $$ (x-1) y^{\prime \prime}-x y^{\prime}+y=0, y(0)=-2, y^{\prime}(0)=6 $$

The differential equation $$ \left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\alpha^{2} y=0 $$ where \(\alpha\) is a parameter, is known as Chebyshev's equation after the Russian mathematician Pafnuty Chebyshev (18211894). Find the general solution \(y(x)=c_{0} y_{1}(x)+c_{1} y_{2}(x)\) of the equation, where \(y_{1}(x)\) and \(y_{2}(x)\) are power series solutions centered at the ordinary point 0 and containing only even powers of \(x\) and odd powers of \(x\), respectively.

$$ \int_{0}^{x} r J_{0}(r) d r=x J_{1}(x) $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about \(x=0 .\) Use \((21)\) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on the interval \((0, \infty)\). $$ x y^{\prime \prime}+y^{\prime}+y=0 $$

Use the power series method to solve the given initial-value problem. $$ (x+1) y^{\prime \prime}-(2-x) y^{\prime}+y=0, y(0)=2, y^{\prime}(0)=-1 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the recurrence relation found by the method of Frobenius first with the largest root \(r_{1}\). How many solutions did you find? Next use the recurrence relation with the smaller root \(r_{2}\). How many solutions did you find? $$ x y^{\prime \prime}+(x-6) y^{\prime}-3 y=0 $$

Use the power series method to solve the given initial-value problem. $$ y^{\prime \prime}-2 x y^{\prime}+8 y=0, y(0)=3, y^{\prime}(0)=0 $$

\(x=0\) is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the recurrence relation found by the method of Frobenius first with the largest root \(r_{1}\). How many solutions did you find? Next use the recurrence relation with the smaller root \(r_{2}\). How many solutions did you find? $$ x(x-1) y^{\prime \prime}+3 y^{\prime}-2 y=0 $$

Use the power series method to solve the given initial-value problem. $$ \left(x^{2}+1\right) y^{\prime \prime}+2 x y^{\prime}=0, y(0)=0, y^{\prime}(0)=1 $$

Use the change of variables \(s=\frac{2}{\alpha} \sqrt{\frac{k}{m}} e^{-\alpha n / 2}\) to show that the differential equation of the aging spring \(m x^{\prime \prime}+k e^{-\alpha t} x=0\), \(\alpha>0\), becomes $$ s^{2} \frac{d^{2} x}{d s^{2}}+s \frac{d x}{d s}+s^{2} x=0 $$

(a) The differential equation \(x^{4} y^{\prime \prime}+\lambda y=0\) has an irregular singular point at \(x=0\). Show that the substitution \(t=1 / x\) yields the differential equation $$ \frac{d^{2} y}{d t^{2}}+\frac{2}{t} \frac{d y}{d t}+\lambda y=0 $$ which now has a regular singular point at \(t=0\). (b) Use the method of this section to find two series solutions of the second equation in part (a) about the singular point \(t=0\) (c) Express each series solution of the original equation in terms of elementary functions.

Show that \(y=x^{1 / 2} w\left({ }_{3}^{2} \alpha x^{3 / 2}\right)\) is a solution of Airy's differential equation \(y^{\prime \prime}+\alpha^{2} x y=0, x>0\), whenever \(w\) is a solution of Bessel's equation of order \(_{3}^{1}\); that is, \(t^{2} w^{\prime \prime}+t w^{\prime}+\left(t^{2}-\frac{1}{9}\right) w=0\), \(t>0 .\) [Hint: After differentiating, substituting, and simplifying. then let \(\left.t=\frac{2}{3} \alpha x^{3 / 2} .\right]\)

Show that \(y=x^{1 / 2} w\left(\frac{2}{3} \alpha x^{3 / 2}\right)\) is a solution of Airy's differe equation \(y^{\prime \prime}+\alpha^{2} x y=0, x>0\), whenever \(w\) is a solutic Bessel's equation of order \(\frac{1}{3}\); that is, \(t^{2} w^{\prime \prime}+t w^{\prime}+\left(t^{2}-\frac{1}{9}\right) w\) \(t>0\). [Hint: After differentiating, substituting, and simp ing, then let \(t=\frac{2}{3} \alpha x^{3 / 2}\).1

$$ y^{\prime \prime}+e^{x} y^{\prime}-y=0 $$

Without actually solving the differential equation \((\cos x) y^{\prime \prime}+\) \(y^{\prime}+5 y=0\), find a lower bound for the radius of convergence of power series solutions about \(x=0 .\) About \(x=1\).

Discuss how you would define a regular singular point for the linear third- order differential equation $$ a_{3}(x) y^{\prime \prime \prime}+a_{2}(x) y^{\prime \prime}+a_{1}(x) y^{\prime}+a_{0}(x) y=0 $$

Each of the differential equations $$ x^{3} y^{\prime \prime}+y=0 \quad \text { and } \quad x^{2} y^{\prime \prime}+(3 x-1) y^{\prime}+y=0 $$ has an irregular singular point at \(x=0 .\) Determine whether the method of Frobenius yields a seriessolution of each differential equation about \(x=0\). Discuss and explain your findings.

Is \(x=0\) an ordinary or a singular point of the differential equation \(x y^{\prime \prime}+(\sin x) y=0 ?\) Defend your answer with sound mathematics.

We have seen that \(x=0\) is a regular singular point of any Cauchy-Euler equation \(a x^{2} y^{\prime \prime}+b x y^{\prime}+c y=0\). Are the indicial equation (14) for a Cauchy-Euler equation and its auxiliary equation related? Discuss.

For purposes of this problem, ignore the graphs given in Figure 5.1.1. If Airy's \(\mathrm{DE}\) is written as \(y^{\prime \prime}=-x y\), what can we say about the shape of a solution curve if \(x>0\) and \(y>0 ?\) If \(x>0\) and \(y<0 ?\)

(a) Find two power series solutions for \(y^{\prime \prime}+x y^{\prime}+y=0\) and express the solutions \(y_{1}(x)\) and \(y_{2}(x)\) in terms of summation notation. (b) Use a CAS to graph the partial sums \(S_{N}(x)\) for \(y_{1}(x)\). Use \(N=2,3,5,6,8,10 .\) Repeat using the partial sums \(S_{N}(x)\) for \(y_{2}(x)\). (c) Compare the graphs obtained in part (b) with the curve obtained using a numerical solver. Use the initial conditions \(y_{1}(0)=1, y_{1}^{\prime}(0)=0\), and \(y_{2}(0)=0, y_{2}^{\prime}(0)=1\). (d) Rexamine the solution \(y_{1}(x)\) in part (a). Express this series as an elementary function. Then use (5) of Section \(3.2\) to find a second solution of the equation. Verify that this second solution is the same as the power series solution \(y_{2}(x)\).

If \(n\) is an integer, use the substitution \(R(x)=(\alpha x)^{-1 / 2} Z(x)\) to show that the differential equation $$ x^{2} \frac{d^{2} R}{d x^{2}}+2 x \frac{d R}{d x}+\left[\alpha^{2} x^{2}-n(n+1)\right] R=0 $$ becomes $$ x^{2} \frac{d^{2} Z}{d x^{2}}+x \frac{d Z}{d x}+\left[\alpha^{2} x^{2}-\left(n+\frac{1}{2}\right)^{2}\right] Z=0 $$

Show that the differential equation $$ \sin \theta \frac{d^{2} y}{d \theta^{2}}+\cos \theta \frac{d y}{d \theta}+n(n+1)(\sin \theta) y=0 $$ can be transformed into Legendre's equation by means of the substitution \(x=\cos \theta\).

Find the first three positive values of \(\lambda\) for which the proble $$ \begin{aligned} &\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\lambda y=0, \\ &y(0)=0, y(x), y^{\prime}(x) \text { bounded on }[-1,1] \end{aligned} $$ has nontrivial solutions.

Use a root-finding application to find the zeros of \(P_{1}(x)\), \(P_{2}(x), \ldots, P_{7}(x)\). If the Legendre polynomials are built-in functions of your CAS, find the zeros of Legendre polynomials of higher degree. Form a conjecture about the location of the zeros of any Legendre polynomial \(P_{n}(x)\), and then investigate to see whether it is true.